RD Sharma Solutions Class 12 Mathematics Chapter 22 VSA

Access premium articles, webinars, resources to make the best decisions for career, course, exams, scholarships, study abroad and much more with

Plan, Prepare & Make the Best Career Choices

# RD Sharma Solutions Class 12 Mathematics Chapter 22 VSA

Edited By Satyajeet Kumar | Updated on Jan 24, 2022 06:21 PM IST

The RD Sharma mathematics collection is said to be the topmost solution for students to prepare themselves for board examinations. The CBSE students are most likely to get recommended by their teachers to get help from the RD Sharma class 12 solution of Algebra of vectors exercise VSA to guide for better understanding. RD Sharma solutions Therefore, without a doubt, any student can opt for the RD Sharma class 12th exercise VSA for a better understanding of maths.

## Algebra of Vectors Excercise: VSA

Algebra of Vectors Exercise Very Short Answer Question 1

Given: Define Zero vector
Explanation:
A vector whose initial and final point coincident or the length of vector is zero known as zero vector or null vector. The null vector is denoted by $\bar{O}$. Also, the magnitude of zero vector is same.

Algebra of Vectors Exercise Very Short Answer Question 2

Given: Define Unit vector
Explanation:
A unit vector is a vector that has a magnitude of 1. The unit vector in the direction of a vector $\vec{a}$is denoted by $\hat{a}$
$\therefore \mid \hat{a}\mid =1$

Algebra of Vectors Exercise Very Short Answer Question 3

Given: Define position vector of a point
Explanation:
$\Rightarrow$The position vector is said to be a straight line having one end fixed to a body like origin and the second end that is attached to a moving point and this is used to describe the position of the point relative to the body.
$\Rightarrow$A point O is fixed as origin in space (or plane) and P is any point, then
$\bar{OP}$ is called a position vector of P w.r.t O

Algebra of Vectors Exercise Very Short Answer Question 4

Hint: You must know the rules of vector functions.
Given: Write $\overrightarrow{P Q}+\overrightarrow{R P}+\overrightarrow{Q R}$ in simplest form
Solution: We have,
$\begin{array}{ll} \overrightarrow{P Q}+\overrightarrow{R P}+\overline{Q R} \\ \end{array}$
$\begin{array}{ll} \Rightarrow \overline{P Q}+\overline{Q R}+\overline{R P} \end{array}$ $\begin{array}{ll} \quad & {[\overline{P R}=\overline{P Q}+\overline{Q R}]} \\ \end{array}$
$\begin{array}{ll} \Rightarrow \overline{P R}+\overline{R P} \end{array}$ $\begin{array}{ll} \quad & {[\overline{P R}=-\overline{R P}]} \end{array}$ because of change in direction

Hence $\Rightarrow \bar{0}$

Algebra of Vectors Exercise Very Short Answer Question 5

Answer:$X=0,Y=0$
Hint: You must know the rules of vector functions.
Given: If $\vec{a}$ and $\vec{b}$are two non-collinear vectors, such that $x\vec{a}+y\vec{b}=\vec{0}$, then write the values of $x$ and $y$
Solution: We have, $\vec{a}$ and $\vec{b}$ are non-collinear vectors and $x\vec{a}+y\vec{b}=\vec{0}$
$\Rightarrow \frac{a}{b}=-\frac{y}{x}$
Given that $\vec{a}$ and $\vec{b}$ are non collinear , so $\vec{a}$ and $\vec{b}$ should not be parallel ,
Therefore, $\frac{y}{x}=0$
$\Rightarrow y=0$
Similarly , the given equation can also be written as $\frac{b}{a}=-\frac{x}{y}$
Given that $\vec{a}$ and $\vec{b}$ are non collinear , so $\vec{a}$ and $\vec{b}$ should not be parallel ,
Therefore, $\frac{x}{y}=0$
$\Rightarrow x=0$
Hence we get , $x=0$ and $y=0$

Algebra of Vectors Exercise Very Short Answer Question 6

Answer: $\vec{a}+\vec{b},\vec{a}-\vec{b}$
Hints: You must know the rules of vector functions.
Given: If $\vec{a}$ and $\vec{b}$represents two adjacent sides of a parallelogram, then write vectors representing its diagonals.
Solution:

Let $\vec{a}$ and $\vec{b}$ represents two adjacent sides of a parallelogram ABCD.
∴ AB = DC and AD = BC [Because diagonals of parallelogram is equal]
\begin{aligned} &\Rightarrow \overrightarrow{D C}=\overline{A B}=\vec{a} \\ &\overrightarrow{A D}=\overrightarrow{B C}=\vec{b} \end{aligned}
In ABC
\begin{aligned} &\overrightarrow{A B}+\overrightarrow{B C}=\overrightarrow{A C} \\ &\vec{a}+\vec{b}=\overrightarrow{A C} \end{aligned}
Now, In ABD
\begin{aligned} & \overrightarrow{A D}+\overline{D B}=\overline{A B} \\ & \vec{b}+\overrightarrow{D B}=\vec{a} \\ \Rightarrow \overrightarrow{D B} &=\vec{a}-\vec{b} \end{aligned}
Vectors representing its diagonals are $\left ( \vec{a}+\vec{b} \right ),\left ( \vec{a}-\vec{b} \right )$

Algebra of Vectors Exercise Very Short Answer Question 7

Answer: $\vec{0}$
Hint: You must know the rules of vector functions.
Given: If $\vec{a},\vec{b},\vec{c}$ represents the sides of a triangle taken in order, then write the value of $\vec{a}+\vec{b}+\vec{c}$
Solution: Let ABC be a triangle, such that $\overrightarrow{B C}=\vec{a}, \overline{C A}=\vec{b} \& \overrightarrow{A B}=\vec{c}$
Then,
\begin{aligned} \vec{a}+\vec{b}+\vec{c}=\overrightarrow{B C}+\overline{C A}+\overline{A B} \\ \end{aligned}
\begin{aligned} &=\overrightarrow{B A}+\overrightarrow{A B} \\ \end{aligned} \begin{aligned} \left [ we\; can \; write \vec{BC}+\vec{CA}=\vec{BA} \right ] \end{aligned}
\begin{aligned} \text { But } \overrightarrow{A B}=-\overrightarrow{B A} \\ \end{aligned} $[Because \; of \; opposite\; direction]$
\begin{aligned} \therefore \overrightarrow{B A}+\overrightarrow{A B} \Rightarrow \overrightarrow{0} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Question 8

Answer: $\vec{0}$
Hint: You must know the rules of vector functions.
Given: If $\vec{a},\vec{b},\vec{c}$ are position vectors of vertices A, B and C respectively of a triangle ABC, write the value of $\vec{AB},\vec{BC},\vec{CA}$
Solution: Given $\vec{a},\vec{b},\vec{c}$ are position vectors of vertices A, B and C respectively
Then,
\begin{aligned} &\overrightarrow{A B}=\vec{b}-\vec{a} \\ &\overrightarrow{B C}=\vec{c}-\vec{b} \\ &\overrightarrow{C A}=\vec{a}-\vec{c} \end{aligned}

Consider,
$\vec{AB}+\vec{BC}+\vec{CA}=\vec{b}-\vec{a}+\vec{c}-\vec{b}+\vec{a}-\vec{c}$
$\Rightarrow \vec{0}$

Algebra of Vectors Exercise Very Short Answer Question 9

Answer: $2\left ( \vec{c}-\vec{a} \right )$
Hint: You must know the rules of vector functions.
Given: If $\vec{a},\vec{b},\vec{c}$ are position vectors of points A, B and C respectively, write the value of $\vec{AB},\vec{BC},\vec{AC}$
Solution: $\vec{a},\vec{b},\vec{c}$are position vectors of points A, B and C respectively.
Then,
\begin{aligned} &\overrightarrow{A B}=\vec{b}-\vec{a} \\ &\overrightarrow{B C}=\vec{c}-\vec{b} \quad \text { because }[\overline{C A}=\vec{a}-\vec{c}=-\overrightarrow{A C}=\vec{c}-\vec{a}] \text { Opposite direction } \\ &\overline{C A}=\vec{c}-\vec{a} \end{aligned}
Therefore,
\begin{aligned} \overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{A C} &=\vec{b}-\vec{a}+\vec{c}-\vec{b}+\vec{c}-\vec{a} \\ &=2 \vec{c}-2 \vec{a} \\ & \Rightarrow 2(\vec{c}-\vec{a}) \end{aligned}

Algebra of Vectors Exercise Very Short Answer Question 10

Answer: $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$
Hint: You must know the rules of vector functions
Given: If $\vec{a}+\vec{b}+\vec{c}$are position vectors of vertices of a triangle, then write the position vector of centroid.
Solution: Let ABC be a triangle and D,E and F are the mid-points of sides BC, CA and AB respectively.
Also, Let $\vec{a}+\vec{b}+\vec{c}$are position vectors of A, B, C respectively.
Then position vectors of D, E and F are
$\left(\frac{\vec{b}+\vec{c}}{2}\right) \cdot\left(\frac{\vec{c}+\vec{a}}{2}\right),\left(\frac{\vec{a}+\vec{b}}{2}\right)$ respectively.
The position vector of a point divides AD in the ratio of 2 is

$\frac{1 \cdot \vec{a}+2\left(\frac{\vec{b}+\vec{c}}{2}\right)}{3}=\frac{\vec{a}+\vec{b}+\vec{c}}{3}$

Similarly, position vectors of the points divides BE, CF in the ratio of 2:1 are equal to

$\frac{\vec{a}+\vec{b}+\vec{c}}{3}$

Thus, the points dividing AD in ratio 2:1 also divides BE, CF in the ratio.

Hence, medians of triangle are concurrent and the position of centroid is $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$

Algebra of Vectors Exercise Very Short Answer Type Question 11

Answer: $\vec{0}$
Hint: You must know the rules of vector functions
Given: If G is denotes the centroid of ?ABC , then write the value of $\vec{GA}+\vec{GB}+\vec{GC}$
Solution: Let $\vec{a},\vec{b},\vec{c}$be the position vectors of the vertices A, B and C respectively.
Then, the position of centroid G is $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$
Thus,
\begin{aligned} \overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C} &=\vec{a}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{3}\right)+\vec{b}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{3}\right)+\vec{c}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{3}\right) \\ & \Rightarrow(\vec{a}+\vec{b}+\vec{c})-3\left(\frac{\vec{a}+\vec{b}+\vec{c}}{3}\right) \\ & \Rightarrow(\vec{a}+\vec{b}+\vec{c})-(\vec{a}+\vec{b}+\vec{c}) \\ & \Rightarrow \overrightarrow{0} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 12

Answer: $\frac{1}{3}\left ( 2\vec{b}+\vec{a} \right )$
Hint: You must know the rules of vector functions
Given: If $\vec{a}$ & $\vec{b}$denote the position vectors of points A and B respectively and C is appoint on AB, such that 3AC = 2AB, then write the position vector of C
Solution: Let $\vec{c}$is the position vector of c
Now,
$\vec{AB}=\vec{b}-\vec{a}$
$\vec{AC}=\vec{c}-\vec{a}$

Consider,

\begin{aligned} &3 \overrightarrow{A C}=2 \overrightarrow{A B} \\ &3(\vec{c}-\vec{a})=2(\vec{b}-\vec{a}) \\ &3 \vec{c}-3 \vec{a}=2 \vec{b}-2 \vec{a} \\ &3 \vec{c}=2 \vec{b}-2 \vec{a}+3 \vec{a} \\ &3 \vec{c}=2 \vec{b}+\vec{a} \\ &\vec{c}=\frac{1}{3}(2 \vec{b}+\vec{a}) \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 13

Answer: $\lambda =2$
Hint: You must know the rules of vector functions
Given: If D is the mid-point of sides BC of a triangle ABC such $\vec{AB}+\vec{AC}=\lambda \vec{AD}$ find $\lambda$
Solution: D is mid-point of side BC of a ABC, $\vec{AB}+\vec{AC}=\lambda \vec{AD}$
Let $\vec{a},\vec{b},\vec{c}$is a position vectors of AB, BC, CA
Now, the position vector of D is $\frac{\vec{b}+\vec{c}}{2}$
Then,
$\vec{AB}=\vec{b}-\vec{a}$
$\vec{AC}=\vec{c}-\vec{a}$
$\begin{gathered} \overrightarrow{A D}=\frac{\vec{b}+\vec{c}}{2}-\vec{a} \\ \therefore \overrightarrow{A B}+\overline{A C}=\lambda \overrightarrow{A D} \\ (\vec{b}-\vec{a})+(\vec{c}-\vec{a})=\lambda\left(\frac{\vec{b}+\vec{c}-2 \vec{a}}{2}\right) \\ \vec{b}-\vec{a}+\vec{c}-\vec{a}=\lambda\left(\frac{\vec{b}+\vec{c}-2 \vec{a}}{2}\right) \\ (\vec{b}+\vec{c}-2 \vec{a})\left(\frac{2}{\vec{b}+\vec{c}-2 \vec{a}}\right)=\lambda \\ \therefore \lambda=2 \end{gathered}$

Algebra of Vectors Exercise Very Short Answer Type Question 14

Answer: $\vec{0}$
Hint: You must know the rules of vector functions
Given: If D,E, F are the mid-points of sides BC,CA and AB respectively of $\Delta ABC$
Write the value of $\vec{AD}+\vec{BE}+\vec{CF}$
Solution: D, E, F are the mid-points of sides BC, CA, AB respectively
Then, the position vectors of the mid-points
D, E, F are given by $\frac{\vec{b}+\vec{c}}{2},\frac{\vec{c}+\vec{a}}{2},\frac{\vec{a}+\vec{b}}{2}$
Now,
\begin{aligned} &\overrightarrow{A D}+\overrightarrow{B E}+\overline{C F} \\ &\left(\frac{\vec{b}+\vec{c}}{2}\right)-\vec{a}+\left(\frac{\vec{c}+\vec{a}}{2}\right)-\vec{b}+\left(\frac{\vec{a}+\vec{b}}{2}\right)-\vec{c} \\ &2\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)-(\vec{a}+\vec{b}+\vec{c}) \\ &\Rightarrow(\vec{a}+\vec{b}+\vec{c})-(\vec{a}+\vec{b}+\vec{c}) \\ &\Rightarrow \overrightarrow{0} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 15

Answer: $m=\pm \frac{1}{a}$
Hint: You must know the rules of vector functions
Given: If $\vec{a}$ is a non-zero vector of modulus a and m is a non-zero scalar such that $m\vec{a}$ is a unit vector, find m.
Solution: $\vec{a}$is non-zero vector with modulus a and m
Also $m\vec{a}$ is unit vector
Therefore,
\begin{aligned} &\qquad\mid m \vec{a} \mid=1 \\ \end{aligned}
\begin{aligned} &\Rightarrow|m||\vec{a}|=1 \\ \end{aligned}
\begin{aligned} \Rightarrow|m| a=1 \\ \end{aligned}
\begin{aligned} |m|=\frac{1}{a} \\ \end{aligned}
\begin{aligned} &\Rightarrow m=\pm \frac{1}{a} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 16

Answer: $\vec{0}$
Hint: You must know the rules of vector functions
Given: If $\vec{a},\vec{b},\vec{c}$ are the position vectors of the vertices of equilateral triangle. Write value of $\vec{a}+\vec{b}+\vec{c}$
Solution: Let ABC be a given equilateral triangle and vertices are $A\vec{\left ( a \right )},B\vec{\left ( b \right )},C\vec{\left ( c \right )}$. Also $O\vec{\left ( o \right )},$ be your orthocentre.
We know the centroid and orthocenter of equilateral triangles coincide at a point.
Orthocentre of $\Delta ABC=0$
Centroid of $\Delta ABC=\vec{0}$
$\Rightarrow \frac{\vec{a}+\vec{b}+\vec{c}}{3}=\vec{0}$
$\vec{a}+\vec{b}+\vec{c}=\vec{0}$

Algebra of Vectors Exercise Very Short Answer Type Question 17

A nswer: $\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}+\frac{1}{\sqrt{3}} \hat{k}$
Hint: You must know the rules of vector functions
Given: Write a unit vector making equal acute angles with the coordinate axes
Solution: Suppose $\vec{r}$ makes an angle $\alpha$ with each of the axes OX, OY and OZ
Then its direction cosines are $l=\cos \alpha, m=\cos \alpha, n=\cos \alpha$
Now,
\begin{aligned} &l^{2}+m^{2}+n^{2}=1 \\ &\Rightarrow l^{2}+l^{2}+l^{2}=1 \; \; \; \; \; \; \; \; \; \; \; \; \quad[l=m=n] \\ &\Rightarrow 3 l^{2}=1 \\ &\Rightarrow l^{2}=\frac{1}{3} \\ &\Rightarrow l=\pm \frac{1}{\sqrt{3}} \end{aligned}
Since, we know angle is acute, Hence we only take positive values
∴ Unit vector is $\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}+\frac{1}{\sqrt{3}} \hat{k}$

Algebra of Vectors Exercise Very Short Answer Type Question 18

Hint: You must know the rules of vector functions
Given: If a vector makes angles with $\alpha ,\beta$ &$\gamma$ with OX, OY and OZ respectively, Write $\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma$
Solution: Suppose, a vector $\vec{OP}$makes angles $\alpha ,\beta$ &$\gamma$ with OX, OY and OZ respectively
Then direction cosines of vectors are given by $l=\cos \alpha, m=\cos \beta, n=\cos \gamma$
Consider,
$\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=\left(1-\cos ^{2} \alpha\right)+\left(1-\cos ^{2} \beta\right)+\left(1-\cos ^{2} \gamma\right)$
\begin{aligned} &=3-\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right)\\ &=3-\left(l^{2}+m^{2}+n^{2}\right)\\ &=3-1 \quad\left[l^{2}+m^{2}+n^{2}=1\right]\\ &=2 \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 19

Answer: $6(\sqrt{2} \hat{i}+\hat{j}-\hat{k})$
Hint: You must know the rules of vector functions
Given: Write a vector of magnitude 12 units which makes $45^{0}$ angles with x-axis, $60^{0}$angle with y-axis, obtuse angle with z- axis.
Solution: Suppose a vector $\vec{r}$makes an angle $45^{0}$ with OX, $60^{0}$ with OY and having magnitude 12 units.
$l=\cos 45^{\circ}=\frac{1}{\sqrt{2}} \text { and } m=\cos 60^{\circ}=\frac{1}{2}$
Now,
\begin{aligned} &l^{2}+m^{2}+n^{2}=1 \\ &\frac{1}{2}+\frac{1}{4}+n^{2}=1 \\ &n^{2}=1-\frac{1}{2}-\frac{1}{4} \end{aligned}
\begin{aligned} &n^{2}=\frac{1}{4} \end{aligned}
\begin{aligned} &n^{2}=\pm \frac{1}{4} \end{aligned}
But angle obtuse angle along z-axis, so we use negative value.
\begin{aligned} &n^{2}= \frac{-1}{4} \end{aligned}
Therefore,
\begin{aligned} &\vec{r}=|\vec{r}|(\hat{i}+m \hat{j}+n \hat{k}) \\ &=12\left(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{2} \hat{j}-\frac{1}{2} \hat{k}\right) \\ &=6(\sqrt{2} \hat{i}+\hat{j}-\hat{k}) \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 20

Hint: You must know the rules of vector functions
Given: Whose projections on coordinate axis are 12, 3, 4 units. Write length of vector
Solution:Projections on coordinate axis are 12, 3, 4 units
Therefore, length of vector
\begin{aligned} &=\sqrt{(12)^{2}+(3)^{2}+(4)^{2}} \\ &=\sqrt{144+9+16} \\ &=\sqrt{169} \\ &=13 \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 21

Answer: $3 \hat{i}+\frac{11}{3} \hat{j}+5 \hat{k}$
Hint: You must know the rules of vector functions
Given: Write the position vector of a point dividing the line segment joining points A and B with position vectors $\vec{a}$ &$\vec{b}$ externally in the ratio 1:4 , where,
\begin{aligned} &\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k} \\ &b=-\hat{i}+\hat{j}+\hat{k} \end{aligned}
Solution: The position vectors of A and B are
\begin{aligned} &\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k} \\ &b=-\hat{i}+\hat{j}+\hat{k} \end{aligned}
Let C divides AB in the ratio such that AB:CB=1:4
Position vector of C=
\begin{aligned} &\frac{1(-\hat{i}+\hat{j}+\hat{k})-4(2 \hat{i}+3 \hat{j}+4 \hat{k})}{1-4} \\ &\Rightarrow \frac{-\hat{i}+\hat{j}+\hat{k}-8 \hat{i}-12 \hat{j}-16 \hat{k}}{-3} \\ &\Rightarrow \frac{-9 \hat{i}-11 \hat{j}-15 \hat{k}}{-3} \Rightarrow 3 \hat{i}+\frac{11 \hat{j}}{3}+5 \hat{k} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 22

Answer: $\frac{6}{7},\frac{-2}{7},\frac{3}{7}$
Hint: You must know the rules of vector functions
Solution: $\vec{r}=6\hat{i}-2\hat{j}+3\hat{k}$
The direction cosines are
\begin{aligned} &\frac{6}{\sqrt{(6)^{2}+(-2)^{2}+(3)^{2}}}, \frac{-2}{\sqrt{(6)^{2}+(-2)^{2}+(3)^{2}}}, \frac{3}{\sqrt{(6)^{2}+(-2)^{2}+(3)^{2}}} \\ &\begin{array}{l} \end{array} \end{aligned}
$\Rightarrow \frac{6}{\sqrt{49}}, \frac{-2}{\sqrt{49}}, \frac{3}{\sqrt{49}} \\$
$\Rightarrow \frac{6}{7}, \frac{-2}{7}, \frac{3}{7}$

Algebra of Vectors Exercise Very Short Answer Type Question 23

Answer:$\frac{-\hat{i}+2\hat{j}-\hat{k}}{\sqrt{6}}$
Hint: You must know the rules of vector functions
Given: $\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{j}+\hat{k} \& \vec{c}=\hat{k}+\hat{i}$ find unit vector parallel to $\vec{a}+\vec{b}-2\vec{c}$
Solution: $\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{j}+\hat{k} \& \vec{c}=\hat{k}+\hat{i}$
Now,
\begin{aligned} &\vec{a}+\vec{b}-2 \vec{c}=\hat{i}+\hat{j}+\hat{j}+\hat{k}-2 \hat{k}-2 \hat{i} \\ &=-\hat{i}+2 \hat{j}-\hat{k} \end{aligned}
Unit vector parallel to,
\begin{aligned} &\vec{a}+\vec{b}-2 \vec{c}=\frac{-\hat{i}+2 \hat{j}-\hat{k}}{\sqrt{(-1)^{2}+(2)^{2}+(1)^{2}}} \\ &=\frac{-\hat{i}+2 \hat{j}-\hat{k}}{\sqrt{6}} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 24

Answer: $\frac{1}{\sqrt{41}}\left ( 3\hat{i}+4\hat{j}-4\hat{k} \right )$
Hint: You must know the rules of vector functions
Given: $\vec{a}=\hat{i}+2\hat{j},\vec{b}=\hat{j}+2\hat{k}$, find unit vector $3\vec{a}-2\vec{b}$
Solution: $\vec{a}=\hat{i}+2\hat{j},\vec{b}=\hat{j}+2\hat{k}$
Hence,
\begin{aligned} &3 \vec{a}-2 \vec{b}=3(\hat{i}+2 \hat{j})-2(\hat{j}+2 \hat{k}) \\ &=3 \hat{i}+6 \hat{j}-2 \hat{j}-4 \hat{k} \\ &=3 \hat{i}+4 \hat{j}-4 \hat{k} \end{aligned}

Hence, unit vector along,

\begin{aligned} &3 \vec{a}-2 \vec{b}=\frac{3 \hat{i}+4 \hat{j}-4 \hat{k}}{\sqrt{(3)^{2}+(4)^{2}+(-4)^{2}}} \\ &=\frac{3 \hat{i}+4 \hat{j}-4 \hat{k}}{\sqrt{9+16+16}} \\ &=\frac{1}{\sqrt{41}}(3 \hat{i}+4 \hat{j}-4 \hat{k}) \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 25

Answer: $-\hat{i}+5\hat{j}-12\hat{k}$
Hint: You must know the rules of vector functions
Given: Find position vector of point dividing line segment
$\hat{i}+\hat{j}-2\hat{k},2\hat{i}-\hat{j}+3\hat{k}$ Externally in 2:3
Solution: Let A and B be the points with vectors
\begin{aligned} &\vec{a}=\hat{i}+\hat{j}-2 \hat{k} \\ &\vec{b}=2 \hat{i}-\hat{j}+3 \hat{k} \end{aligned}respectively
Let C divide AB externally with ratio 2:3 such AC: CB=2:3
Position vector of C=
\begin{aligned} &=\frac{2(2 \hat{i}-\hat{j}+3 \hat{k})-3(\hat{i}+\hat{j}-2 \hat{k})}{2-3} \\ &=\frac{(4 \hat{i}-2 \hat{j}+6 \hat{k})-3 \hat{i}-3 \hat{j}+6 \hat{k}}{-1} \\ &=\frac{1}{-1}(\hat{i}-5 \hat{j}+12 \hat{k}) \\ &=-\hat{i}+5 \hat{j}-12 \hat{k} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 26

Answer: $\frac{1}{\sqrt{3}}\left ( \hat{i}+\hat{j}+\hat{k} \right )$
Hint: You must know the rules of vector functions
Given:$\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{j}+\hat{k}, \vec{c}=\hat{k}+\hat{i}$ find unit vector in the direction of $\vec{a}+\vec{b}+\vec{c}$
Solution:$\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{j}+\hat{k}, \vec{c}=\hat{k}+\hat{i}$
Then,
\begin{aligned} &\vec{a}+\vec{b}+\vec{c}=\hat{i}+\hat{j}+\hat{j}+\hat{k}+\hat{k}+\hat{i} \\ &=2(\hat{i}+\hat{j}+\hat{k}) \\ &\therefore|\vec{a}+\vec{b}+\vec{c}|=\sqrt{2^{2}+2^{2}+2^{2}} \\ &=\sqrt{4+4+4} \\ &=\sqrt{12} \\ &=2 \sqrt{3} \end{aligned}
Therefore, unit vector in the direction of

$\vec{a}+\vec{b}+\vec{c}=\frac{2(\hat{i}+\hat{j}+\hat{k})}{2 \sqrt{3}}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

Algebra of Vectors Exercise Very Short Answer Type Question 27

Answer: $\sqrt{398}$
Hint: You must know the rules of vector functions
Given: If
\begin{aligned} &\vec{a}=3 \hat{i}-\hat{j}-4 \hat{k} \\ &\vec{b}=-2 \hat{i}+4 \hat{j}-3 \hat{k}, \text { find }|3 \vec{a}-2 \vec{b}+4 \vec{c}| \\ &\vec{c}=\hat{i}+2 \hat{j}-\hat{k} \end{aligned}
Solution:
\begin{aligned} &\vec{a}=3 \hat{i}-\hat{j}-4 \hat{k} \\ &\vec{b}=-2 \hat{i}+4 \hat{j}-3 \hat{k}, \\ &\vec{c}=\hat{i}+2 \hat{j}-\hat{k} \end{aligned}
Now,
\begin{aligned} &3 \vec{a}-2 \vec{b}+4 \vec{c}=3(3 \hat{i}-\hat{j}-4 \hat{k})-2(-2 \hat{i}+4 \hat{j}-3 \hat{k})+4(\hat{i}+2 \hat{j}-\hat{k}) \\ &=9 \hat{i}-3 \hat{j}-12 \hat{k}+4 \hat{i}-8 \hat{j}+6 \hat{k}+4 \hat{i}+8 \hat{j}-4 \hat{k} \\ &=17 \hat{i}-3 \hat{j}-10 \hat{k} \\ \end{aligned}
Hence, \begin{aligned} &\therefore|3 \vec{a}-2 \vec{b}+4 \vec{c}|=\sqrt{(17)^{2}-3^{2}-(10)^{2}} \\ \end{aligned}
\begin{aligned} &=\sqrt{289+9+100} \\ &=\sqrt{398} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 28

Answer: $\theta =30^{o}$
Hint: You must know the rules of vector functions
Given: A unit vector $\vec{r}$make angles $\frac{\pi }{3},\frac{\pi }{2}$ with $\hat{j}$ &$\hat{k}$ and an acute angle $\theta$ with $\hat{i}$.find $\theta$
Solution: Angle $\frac{\pi }{3},\frac{\pi }{2}$ with $\hat{j}$ &$\hat{k}$
Let l , m , n be the direction cosines
\begin{aligned} &l=\cos \theta, m=\cos \frac{\pi}{3}, n=\cos \frac{\pi}{2} \\ &l=\cos \theta, m=\frac{1}{2}, n=0 \\ \end{aligned}
Now,\begin{aligned} &l^{2}+m^{2}+n^{2}=1 \\ \end{aligned}
\begin{aligned} &l^{2}+\frac{1}{4}+0=1 \\ &l^{2}=\frac{3}{4} \\ &l=\pm \sqrt{\frac{3}{4}} \Rightarrow \pm \frac{\sqrt{3}}{2} \\ \end{aligned}

But we know angle made with $\hat{i}$is an acute angle so, we use the positive value.

$\therefore \theta =30^{o}$ $\left [ \therefore \cos ^{-1}\left ( \frac{\sqrt{3}}{2} \right )=30^{0} \right ]$

Algebra of Vectors Exercise Very Short Answer Type Question 29

Answer: $\frac{3}{7}\hat{i}-\frac{2}{7}\hat{j}+\frac{6}{7}\hat{k}$
Hint: You must know the rules of vector functions
Given: Find unit vector in direction of $\hat{a}=3\hat{i}-2\hat{j}+6\hat{k}$
Solution: We have,
$\hat{a}=3\hat{i}-2\hat{j}+6\hat{k}$
\begin{aligned} &|\vec{a}|=\sqrt{(3)^{2}+(-2)^{2}+(6)^{2}} \Rightarrow \sqrt{49}=7 \\ \end{aligned}
Unit vector in direction of $\hat{a}$ is
\begin{aligned} &\hat{a}=\frac{\vec{a}}{|\vec{a}|}=\frac{1}{7}(3 \hat{i}-2 \hat{j}+6 \hat{k}) \\ &\Rightarrow \frac{3}{7} \hat{i}-\frac{2}{7} \hat{j}+\frac{6}{7} \hat{k} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 30

Answer: $\frac{1}{3}\left ( \hat{i}+2\hat{j} +2\hat{k}\right )$
Hint: You must know the rules of vector functions
Given: $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$and $\vec{b}=2\hat{i}+4\hat{j}+9\hat{k}$find vector parallel to $\vec{a}+\vec{b}$
Solution:
$\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$
$\vec{b}=2\hat{i}+4\hat{j}+9\hat{k}$
Now,\begin{aligned} &\vec{a}+\vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}+2 \hat{i}+4 \hat{j}+9 \hat{k} \\ \end{aligned}
\begin{aligned} &\vec{a}+\vec{b}=3 \hat{i}+6 \hat{j}+6 \hat{k} \\ \end{aligned}
$=3(\hat{i}+2 \hat{j}+2 \hat{k}) \\$
$|\vec{a}+\vec{b}|=\sqrt{(3)^{2}+(6)^{2}+(6)^{2}} \\$
$=\sqrt{9+36+36} \\$
$=\sqrt{81} \\$
$=9 \\$

Vector parallel to $\vec{a}+\vec{b}$

$\frac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}=\frac{3(\hat{i}+2 \hat{j}+2 \hat{k})}{9}=\frac{1}{3}(\hat{i}+2 \hat{j}+2 \hat{k})$

Algebra of Vectors Exercise Very Short Answer Type Question 31

Answer: $\frac{2}{3}\hat{i}+\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}$
Hint: You must know the rules of vector functions
Given: Find unit vector in direction of $\vec{b}=2\hat{i}+\hat{j}+2\vec{k}$
Solution: $\vec{b}=2\hat{i}+\hat{j}+2\vec{k}$
\begin{aligned} &|\vec{b}|=\sqrt{(2)^{2}+(1)^{2}+(2)^{2}} \\ &=\sqrt{4+1+4} \\ &=\sqrt{9} \\ &=3 \\ \end{aligned}
Unit vector:
\begin{aligned} &\hat{b}=\frac{\vec{b}}{|\vec{b}|}=\frac{1(2 \hat{i}+\hat{j}+2 \hat{k})}{3}=\frac{2}{3} \hat{i}+\frac{1}{3} \hat{j}+\frac{2}{3} \hat{k} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 32

Answer:$\left ( 2,3,1 \right )$
Hint: You must know the rules of vector functions
Given: Find the position vector of mid-point of line AB where $A\left ( 3,4,-2 \right ),B\left ( 1,2,4 \right )$
Solution: $A\left ( 3,4,-2 \right ),B\left ( 1,2,4 \right )$
Let C is the mid-point
∴ Position vector of $C=\frac{\left ( 3\hat{i}+4\hat{j}-2\hat{k} \right )+\left ( \hat{i}+2\hat{j}+4\hat{k} \right )}{2}$
$=\frac{\left ( 4\hat{i}+6\hat{j}+2\hat{k} \right )}{2}$
$\left ( 2\hat{i}+3\hat{j}+\hat{k} \right )$
Hence position vector$=\left ( 2,3,1 \right )$

Algebra of Vectors Exercise Very Short Answer Type Question 33

Answer: $4\hat{i}-2\hat{j}+4\hat{k}$
Hint: You must know the rules of vector functions
Given: Find a vector in direction of $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$, which has magnitude 6 units
Solution:$\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$
\begin{aligned} & \begin{aligned} |\vec{a}|=\sqrt{(2)^{2}+(-1)^{2}+(2)^{2}}=\sqrt{4+1+4} \\ \end{aligned} \end{aligned}
\begin{aligned} & \begin{aligned} &=\sqrt{9} \\ &=3 \\ \end{aligned} \end{aligned}
Vector in direction of
\begin{aligned} & \begin{aligned} \hat{a}=6 \times \frac{\vec{a}}{|\vec{a}|}=\frac{6 \times(2 \hat{i}-\hat{j}+2 \hat{k})}{3} \\ \end{aligned} \end{aligned}
\begin{aligned} & \begin{aligned} =& 2(2 \hat{i}-\hat{j}+2 \hat{k}) \\ =& 4 \hat{i}-2 \hat{j}+4 \hat{k} \end{aligned} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 34

Answer: $\frac{1}{2}$
Hint: You must know the rules of vector functions
Given: What is the cosine of angle with vector $\sqrt{2\hat{i}}+\hat{j}+\hat{k}$ makes with y – axis
Solution: $\sqrt{2\hat{i}}+\hat{j}+\hat{k}$
Direction cosines are
\begin{aligned} &\frac{\sqrt{2}}{\sqrt{(\sqrt{2})^{2}+(1)^{2}+(1)^{2}}}, \frac{1}{\sqrt{(\sqrt{2})^{2}+(1)^{2}+(1)^{2}}}=\frac{1}{\sqrt{(\sqrt{2})^{2}+(1)^{2}+(1)^{2}}} \\ \end{aligned}
\begin{aligned} &\text { OR } \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2} \end{aligned}

Cosine angle along y-axis is $\frac{1}{2}$

Algebra of Vectors Exercise Very Short Answer Type Question 35

Answer: Both have the same magnitude.
Hint: You must know the rules of solving vectors
Given: Write two different vectors having same magnitude
Solution:
Let \begin{aligned} &\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k} \\ &\vec{b}=-2 \hat{i}+\hat{j}-2 \hat{k} \end{aligned}
\begin{aligned} &|\vec{a}|=\sqrt{(2)^{2}+(-1)^{2}+(2)^{2}}=\sqrt{4+1+4}=\sqrt{9}=3 \\ &|\vec{b}|=\sqrt{(-2)^{2}+(1)^{2}+(-2)^{2}}=\sqrt{4+1+4}=\sqrt{9}=3 \end{aligned}

Hence, both vectors are having the same magnitude.

Algebra of Vectors Exercise Very Short Answer Type Question 36

Answer: $\text { Thus, } \vec{a} \text { is parallel to } \vec{b} \text { and hence in the same direction. }$
Hint: You must know the rules of vector functions
Given: Write two different factors having same direction
Solution:
Let $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}, \vec{b}=4 \hat{i}+2 \hat{j}+4 \hat{k}$
The direction cosines of $\hat{a}$ is
\begin{aligned} &l=\frac{2}{\sqrt{(2)^{2}+(1)^{2}+(2)^{2}}}=\frac{2}{\sqrt{9}}=\frac{2}{3} \\ &m=\frac{1}{\sqrt{(2)^{2}+(1)^{2}+(2)^{2}}}=\frac{1}{\sqrt{9}}=\frac{1}{3} \\ &n=\frac{2}{\sqrt{(2)^{2}+(1)^{2}+(2)^{2}}}=\frac{2}{\sqrt{9}}=\frac{2}{3} \\ \end{aligned}

And direction cosines of $\vec{b}$ is
\begin{aligned} &l=\frac{4}{\sqrt{(4)^{2}+(2)^{2}+(4)^{2}}}=\frac{4}{\sqrt{16+4+16}}=\frac{4}{\sqrt{36}}=\frac{4}{6}=\frac{2}{3} \\ &m=\frac{2}{\sqrt{(4)^{2}+(2)^{2}+(4)^{2}}}=\frac{2}{\sqrt{16+4+16}}=\frac{2}{\sqrt{36}}=\frac{2}{6}=\frac{1}{3} \\ &n=\frac{4}{\sqrt{(4)^{2}+(2)^{2}+(4)^{2}}}=\frac{4}{\sqrt{16+4+16}}=\frac{4}{\sqrt{36}}=\frac{4}{6}=\frac{2}{3} \end{aligned}

The direction cosines of $\vec{a}$ and $\vec{b}$ are same.

$\text { Thus, } \vec{a} \text { is parallel to } \vec{b} \text { and hence in the same direction. }$

Algebra of Vectors Exercise Very Short Answer Type Question 37

Answer: $\frac{8}{\sqrt{30}}\left ( 5\hat{i}-\hat{j}+2\hat{k} \right )$
Hint: You must know the rules of vector functions
Given: Write a vector in direction of $5\hat{i}-\hat{j}+2\hat{k}$ having magnitude 8 units
Solution: Let $\vec{a}= 5\hat{i}-\hat{j}+2\hat{k}$
\begin{aligned} &|\vec{a}|=\sqrt{(5)^{2}+(-1)^{2}+(2)^{2}}\\ &=\sqrt{25+1+4}\\ &=\sqrt{30} \end{aligned}
Position vector in the direction of vector is,
\begin{aligned} &=8 \times \frac{\vec{a}}{|\vec{a}|} \\ &=\frac{8}{\sqrt{30}}(5 \hat{i}-\hat{j}+2 \hat{k}) \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 38

Answer: $\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{14}$
Hint: You must know the rules of vector functions
Given: $\hat{i}+2\hat{j}+3\hat{k}$, find direction cosines
Solution: Let $\hat{i}+2\hat{j}+3\hat{k}$
Hence direction cosines are,
\begin{aligned} &\frac{1}{\sqrt{(1)^{2}+(2)^{2}+(3)^{2}}}, \frac{2}{\sqrt{(1)^{2}+(2)^{2}+(3)^{2}}}, \frac{3}{\sqrt{(1)^{2}+(2)^{2}+(3)^{2}}} \\ &=\frac{1}{\sqrt{1+4+9}}, \frac{2}{\sqrt{1+4+9}}, \frac{3}{\sqrt{1+4+9}} \\ &=\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 39

Answer: $\frac{2}{7}\hat{i}-\frac{3}{7}\hat{j}+\frac{6}{7}\hat{k}$
Hint: You must know the rules of vector functions
Given: Find unit vector in direction of $\vec{a}=2\hat{i}-3\hat{j}+6\hat{k}$
Solution:$\vec{a}=2\hat{i}-3\hat{j}+6\hat{k}$
\begin{aligned} &|\vec{a}|=\sqrt{(2)^{2}+(-3)^{2}+(6)^{2}} \\ &=\sqrt{4+9+36} \\ &=\sqrt{49} \\ &=7 \\ \end{aligned}
Unit vector,
\begin{aligned} &\frac{\vec{a}}{|\vec{a}|}=\frac{2 \hat{i}-3 \hat{j}+6 \hat{k}}{7} \\ &=\frac{2}{7} \hat{i}-\frac{3}{7} \hat{j}+\frac{6}{7} \hat{k} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 40

Answer: $-4$
Hint: You must know the rules of vector functions
Given: For what value of ‘a’ the vectors $2\hat{i}-3\hat{j}+4\hat{k}$&$a\hat{i}+6\hat{j}-8\hat{k}$are collinear
Solution: Two vectors are
\begin{aligned} &\vec{p}=2 \hat{i}-3 \hat{j}+4 \hat{k}, q=a \hat{i}+6 \hat{j}-8 \hat{k} \\ \end{aligned}

Vectors are collinear,
\begin{aligned} &\vec{p}=\lambda \vec{q} \\ \end{aligned}
\begin{aligned} &2 \hat{i}-3 \hat{j}+4 \hat{k}=\lambda(\hat{a i}+6 \hat{j}-8 \hat{k}) \\ \end{aligned}
\begin{aligned} &2 \hat{i}-3 \hat{j}+4 \hat{k}=\lambda a \hat{i}+6 \lambda \hat{j}-8 \lambda \hat{k} \\ \end{aligned}

By comparing
\begin{aligned} &-8 \lambda=-3 \Rightarrow \lambda=\frac{-1}{2} \\ \end{aligned}
\begin{aligned} &-6 \lambda=-3 \Rightarrow \lambda=\frac{-1}{2} \\ \end{aligned}
\begin{aligned} &\lambda a=2 \Rightarrow \frac{-1}{2} \times a=2 \Rightarrow a=-4 \end{aligned}
Hence $a=-4$

Algebra of Vectors Exercise very Short Answer type Question 41

Answer: $\frac{-2}{\sqrt{30}},\frac{1}{\sqrt{30}},\frac{-5}{\sqrt{30}}$
Hint: You must know the rules of vector functions
Given: Find direction cosines of $-2\hat{i}+\hat{j}-5\hat{k}$
Solution: Let $\vec{a}=-2\hat{i}+\hat{j}-5\hat{k}$
Then its cosines are,
\begin{aligned} &\frac{-2}{\sqrt{(-2)^{2}+(1)^{2}+(-5)^{2}}}, \frac{1}{\sqrt{(-2)^{2}+(1)^{2}+(-5)^{2}}}, \frac{-5}{\sqrt{(-2)^{2}+(1)^{2}+(-5)^{2}}} \\ &\frac{-2}{\sqrt{4+1+25}}, \frac{1}{\sqrt{4+1+25}}, \frac{-5}{\sqrt{4+1+25}} \\ &\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}} \end{aligned}

Algebra of Vectors Exercise Very Short Answer Type Question 42

Answer: $5\hat{i}-5\hat{j}+3\hat{k}$
Hint: You must know the rules of vector functions
Given: Find sum of $\vec{a}=\hat{i}-2\hat{j},\vec{b}=2\hat{i}-3\hat{j},\vec{c}=2\hat{i}+3\hat{k}$
So, sum of three vectors
\begin{aligned} &\vec{a}+\vec{b}+\vec{c}=\hat{i}-2 \hat{j}+2 \hat{i}-3 \hat{j}+2 \hat{i}+3 \hat{k} \\ &=5 \hat{i}-5 \hat{j}+3 \hat{k} \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 43

Answer: $\frac{3}{7}\hat{i}-\frac{2}{7}\hat{j}+\frac{6}{7}\hat{k}$
Hint: You must know the rules of vector functions
Given: Find unit vector in direction of $\vec{a}=3\hat{i}-2\hat{j}+6\hat{k}$
Solution: $\vec{a}=3\hat{i}-2\hat{j}+6\hat{k}$
Unit vector \begin{aligned} &\frac{\vec{a}}{\mid \vec{a}\mid }=\frac{3 \hat{i}-2 \hat{j}+6 \hat{k}}{\sqrt{(3)^{2}+(-2)^{2}+(6)^{2}}} \\ \end{aligned}
\begin{aligned} &=\frac{3 \hat{\imath}-2 \hat{\jmath}+6 \hat{k}}{\sqrt{9+4+36}} \\ &=\frac{3 \hat{\imath}-2 \hat{\jmath}+6 \hat{k}}{\sqrt{49}} \\ &=\frac{3 \hat{\imath}-2 \hat{\jmath}+6 \hat{k}}{7} \\ &=\frac{3}{7} \hat{\imath}-\frac{2}{7} \hat{\jmath}+\frac{6}{7} \hat{k} \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 44

Answer: $x+y++z=0$
Hint: You must know the rules of vector functions
Given: If $\vec{a}=x \hat{i}+2 \hat{j}-z \hat{k}, \vec{b}=3 \hat{i}-\hat{y} \hat{j}+\hat{k}$are two equal vectors find $x+y++z$
Solution: $\vec{a}=x \hat{i}+2 \hat{j}-z \hat{k}, \vec{b}=3 \hat{i}-\hat{y} \hat{j}+\hat{k}$
They are equal vectors, So, $\vec{a}=\vec{b}$
$x \hat{i}+2 \hat{j}-z \hat{k}=3 \hat{i}-\hat{y} \hat{j}+\hat{k}$
By comparing
$x=3,y=-2,z=-1$
$\therefore x+y+z=3-2-1=0$

Algebra of Vectors Exercise very Short Answer type Question 45

Answer: $\frac{4}{13}\hat{i}+\frac{3}{13}\hat{j}-\frac{12}{13}\hat{k}$
Hint: You must know the rules of vector functions
Given: Find unit vector in direction of sum of vectors
\begin{aligned} &\vec{a}=2 \hat{i}+2 \hat{j}-5 \hat{k}, \\ &\vec{b}=2 \hat{i}+\hat{j}-7 \hat{k} \end{aligned}
Solution: We have,
\begin{aligned} &\vec{a}=2 \hat{i}+2 \hat{j}-5 \hat{k}, \\ &\vec{b}=2 \hat{i}+\hat{j}-7 \hat{k} \end{aligned}
Sum, \begin{aligned} &\vec{p}=\vec{a}+\vec{b}=2 \hat{i}+2 \hat{j}-5 \hat{k}+2 \hat{i}+\hat{j}-7 \hat{k} \\ \end{aligned}
\begin{aligned} &=4 \hat{i}+3 \hat{j}-12 \hat{k} \\ \end{aligned}
∴Required unit vector
\begin{aligned} &\frac{\vec{p}}{|\vec{p}|}=\frac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}=\frac{4 \hat{i}+3 \hat{j}-12 \hat{k}}{\sqrt{(4)^{2}+(3)^{2}+(-12)^{2}}} \\ &=\frac{4 \hat{i}+3 \hat{j}-12 \hat{k}}{\sqrt{16+9+144}} \\ &=\frac{4 \hat{i}+3 \hat{j}-12 \hat{k}}{\sqrt{169}} \\ &=\frac{4 \hat{i}+3 \hat{j}-12 \hat{k}}{13} \\ &\therefore \frac{4}{13} \hat{i}+\frac{3}{13} \hat{j}-\frac{12}{13} \hat{k} \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 46

Answer:$p=\frac{-1}{3}$
Hint: You must know the rules of vector functions
Given: Find value of ‘p’
$3\hat{i}+2\hat{j}+9\hat{k},\hat{i}-2\hat{pj}+3\hat{k}$ are parallel
Solution:
Let $\vec{a}=3\hat{i}+2\hat{j}+9\hat{k}$
$\vec{b}=\hat{i}-2\hat{pj}+3\hat{k}$
If $\vec{a}$ &$\vec{b}$ are parallel
$\vec{b}=\lambda\vec{a}$
\begin{aligned} &\hat{i}-2 \hat{p j}+3 \hat{k}=\lambda(3 \hat{i}+2 \hat{j}+9 \hat{k})\\ &\hat{i}-2 \hat{p j}+3 \hat{k}=3 \lambda \hat{i}+2 \lambda \hat{j}+9 \lambda \hat{k}\\ \end{aligned}
$\therefore$ On comparing,
\begin{aligned} &3 \lambda=1 \Rightarrow \lambda=\frac{1}{3}\\ &3=9 \lambda \Rightarrow \lambda=\frac{1}{3}\\ \end{aligned}
\begin{aligned} &-2 p=2 \lambda \Rightarrow-2 p=2 \times \frac{1}{3} \Rightarrow p=\frac{-1}{3} \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 47

Answer: $\vec{a}=5\left ( \hat{i}+0\hat{j}+\hat{k} \right )$
Hint: You must know the rules of vector functions
Given: Find a vector $\vec{a}$ of magnitude $-5\sqrt{2}$ ,making angle$\frac{\pi }{4}$ with x- axis,$\frac{\pi }{2}$ with y-axis and $\theta$with z-axis
Solution: $\frac{\pi }{4}$ with x- axis, $\frac{\pi }{2}$ with y-axis and $\theta$ with z-axis
\begin{aligned} &l=\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} \\ &m=\cos \frac{\pi}{2}=0 \\ &n=\cos \theta \\ &\therefore l^{2}+m^{2}+n^{2}=1 \\ &\frac{1}{2}+0+\cos ^{2} \theta=1 \\ &\frac{1}{2}=1-\cos ^{2} \theta \\ &\cos ^{2} \theta=1-\frac{1}{2} \\ &\cos ^{2} \theta=\frac{1}{2} \\ &\cos \theta=\frac{1}{\sqrt{2}} \\ \end{aligned}Since $\theta$ is an acute angle
Now,
\begin{aligned} &\vec{a}=|\vec{a}|(l \hat{i}+m \hat{j}+n \hat{k}) \\ &\vec{a}=5 \sqrt{2}\left(\frac{1}{\sqrt{2}} \hat{i}+0 \hat{j}+\frac{1}{\sqrt{2}} \hat{k}\right) \\ &\vec{a}=5(\hat{i}+0 \hat{j}+\hat{k}) \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 48

Answer: $\frac{1}{7}\left ( 3\hat{i}+2\hat{j}+6\hat{k} \right )$
Hint: You must know the rules of vector functions
Given: Find a unit vector in direction of $\vec{PQ}$ when $P\left ( 1,3,0 \right )$ &$Q\left ( 4,5,6 \right )$
Solution: $P\left ( 1,3,0 \right )$ & $Q\left ( 4,5,6 \right )$
\begin{aligned} &\therefore \overrightarrow{P Q}=(4 \hat{i}+5 \hat{j}+6 \hat{k})-(\hat{i}+3 \hat{j}+0 \hat{k}) \\ &=4 \hat{i}+5 \hat{j}+6 \hat{k}-\hat{i}-3 \hat{j}-0 \hat{k} \\ &\overrightarrow{P Q}=3 \hat{i}+2 \hat{j}+6 \hat{k} \\ &|\overrightarrow{P Q}|=\sqrt{(3)^{2}+(2)^{2}+(6)^{2}} \\ &=\sqrt{9+4+36} \\ &=\sqrt{49} \\ &=7 \\ \end{aligned}
Unit Vector:
\begin{aligned} &\frac{\overline{P Q}}{|\overline{P Q}|}=\frac{1}{7}(3 \hat{i}+2 \hat{j}+6 \hat{k}) \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 49

Answer: $6\hat{i}-9\hat{j}+18\hat{k}$
Hint: You must know the rules of vector functions
Given: Find vector in direction of $2\hat{i}-3\hat{j}+6\hat{k}$having magnitude 21 units
Solution: Let $\vec{a}=2\hat{i}-3\hat{j}+6\hat{k}$
Unit vector
\begin{aligned} &\frac{\vec{a}}{|\vec{a}|}=\frac{(2 \hat{i}-3 \hat{j}+6 \hat{k})}{\sqrt{2^{2}+(-3)^{2}+(6)^{2}}}\\ \end{aligned}
\begin{aligned} &=\frac{(2 \hat{i}-3 \hat{j}+6 \hat{k})}{\sqrt{49}}\\ \end{aligned}
\begin{aligned} &=\frac{(2 \hat{i}-3 \hat{j}+6 \hat{k})}{7}\\ \end{aligned}

We know, magnitude is 21 units
\begin{aligned} &|\vec{a}|=21\\ \end{aligned}
\begin{aligned} &\frac{\vec{a}}{21}=\frac{1}{7}(2 \hat{i}-3 \hat{j}+6 \hat{k})\\ \end{aligned}
\begin{aligned} &\vec{a}=3(2 \hat{i}-3 \hat{j}+6 \hat{k})\\ \end{aligned}
\begin{aligned} &=6 \hat{i}-9 \hat{j}+18 \hat{k} \end{aligned}

Algebra of Vectors Exercise very Short Answer type Question 50

Answer:$\left [ -12,8 \right ]$
Hint: You must know the rules of vector functions
Given: If $\mid \vec{a}\mid =4$ and $-3\leq \lambda \leq 2$, Writer range of $\mid \lambda \vec{a}\mid$
Solution: $-3\leq \lambda \leq 2$
$\begin{gathered} \Rightarrow-3 \times|\vec{a}| \leq \lambda|\vec{a}| \leq 2 \times|\vec{a}| \\ \end{gathered}$
Value of $\begin{gathered} |\vec{a}|=4 \\ \end{gathered}$
$\begin{gathered} \therefore \Rightarrow-3 \times 4 \leq \lambda|\vec{a}| \leq 2 \times 4 \\ \end{gathered}$
$\begin{gathered} \Rightarrow-12 \leq \lambda|\vec{a}| \leq 8 \\ \end{gathered}$
Range of $\begin{gathered} \lambda|\vec{a}|=>[-12,8] \end{gathered}$

Algebra of Vectors Exercise very Short Answer type Question 51

Answer:$2\vec{b}-\vec{a}$
Hint: You must know the rules of vector functions
Given: $\Delta DAC$ , if B is mid-point of AC and $\vec{OA}=\vec{a}$,$\vec{OB}=\vec{b}$find $\vec{OC}$
Solution:

$\triangle D A C \overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b}$
$\mathrm{B} \: \: is \: \: mid-point$
$\therefore Position vector of B=\frac{\text { Position vector of A+Position vector of } \mathrm{C}}{2}$
$\overrightarrow{O B}=\frac{\overline{O A}+\overrightarrow{O C}}{2}$
$\vec{b}=\frac{\vec{a}+\overrightarrow{O C}}{2}$
$2 \vec{b}=\vec{a}+\overrightarrow{O C}$
$\overrightarrow{O C}=2 \vec{b}-\vec{a}$

Answer:$\frac{7}{3}\vec{a}+\frac{4}{3}\vec{b}$
Hint: You must know the rules of vector functions
Given: Write the position vector of the point which divides the join of points with position vectors $3\vec{a}-2\vec{b}$ &$2\vec{a}+3\vec{b}$ in ratio 2:1
Solution: Let R be the point which divides the line joining point with vectors.
$3\vec{a}-2\vec{b}$&$2\vec{a}+3\vec{b}$ in ratio 2:1
And
$\overrightarrow{O A}=3 \vec{a}-2 \vec{b}$
$\overrightarrow{O B}=2 \vec{a}+3 \vec{b}$
Here $m: n=2: 1$
Position vector of $\overline{O R}$is as follows
$\overline{O R}=\frac{m \overline{O B}+n \overline{O A}}{m+n}$
$=\frac{2(2 \vec{a}+3 \vec{b})+1(3 \vec{a}-2 \vec{b})}{2+1}$
$=\frac{4 \vec{a}+6 \vec{b}+3 \vec{a}-2 \vec{b}}{3}$
$=\frac{7 \vec{a}+4 \vec{b}}{3}$
$=\frac{7}{3} \vec{a}+\frac{4}{3} \vec{b}$

Class 12 RD Sharma chapter 22 exercise VSA solution deals with the chapter of Algebra of vectors, which might be a complex chapter for some students to solve. So RD Sharma class 12th exercise VSA provides you with very short answer type questions to take a self-test and evaluate your performance accordingly. The RD Sharma class 12 solutions chapter 22 exercise VSA consists of a total of 52 questions that covers all the essential concepts of the chapter mentioned below-

• Zero vector

• Unit vector

• Position vector

• Vectors in simplified form

• Centroid

• Direction Cosines

• Vectors with same magnitude and direction

Listed below are a few reasons why the RD Sharma class 12th exercise VSA is helpful in the preparation of exams:-

• The questions have a good proportion chance to be asked in the board exams because the exercises designed in the RD Sharma solution cover up each and every concept of every chapter

• The RD Sharma class 12th exercise VSA consists of questions that are frequently asked in board exams so students have to go through each and every exercise thoroughly.

• The solution also helps in solving the homework as it provides you with solved questions and examples and also because teachers use the solutions to assign homework.

• The RD Sharma class 12 chapter 22 exercise VSA is trusted by many students across the country as they have seen that a good practice of the RD Sharma solution has helped them to score big.

• The best part about RD Sharma class 12th exercise VSA is students can download the solutions free of cost from the Career360 website.

## RD Sharma Chapter wise Solutions

1. What are the advantages of this material?

This material contains solutions that act as a guide for students to help them get more knowledge and score good marks in exams.

2. Can I prepare only using this material?

As this material covers the entire syllabus students can prepare from it.

3. Are there any additional charges?

4. Who can use this material?

CBSE students who want to gain more insight on the subject can use this material.

5. Does it cover the entire syllabus?

Yes, this material covers the entire syllabus

## Upcoming School Exams

#### All India Sainik Schools Entrance Examination

Application Date:06 November,2023 - 15 December,2023

#### National Institute of Open Schooling 12th Examination

Application Date:20 November,2023 - 19 December,2023

#### National Institute of Open Schooling 10th examination

Application Date:20 November,2023 - 19 December,2023

#### West Bengal Board 12th Examination

Exam Date:30 November,2023 - 14 December,2023

Exam Date:04 December,2023 - 04 December,2023

Get answers from students and experts

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

4 Jobs Available
##### Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.

4 Jobs Available
##### Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

3 Jobs Available
##### Database Architect

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi

3 Jobs Available
##### Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
##### Geothermal Engineer

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

3 Jobs Available
##### Geotechnical engineer

The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.

The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.

3 Jobs Available
##### Cartographer

How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.

3 Jobs Available
##### Bank Probationary Officer (PO)

A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as  Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.

3 Jobs Available
##### Operations Manager

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

3 Jobs Available
##### Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
##### Finance Executive

A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.

3 Jobs Available
##### Investment Banker

An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.

3 Jobs Available
##### Bank Branch Manager

Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.

3 Jobs Available
##### Treasurer

Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Transportation Planner

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

3 Jobs Available
##### Conservation Architect

A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

2 Jobs Available
##### Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

2 Jobs Available
##### Architect

Individuals in the architecture career are the building designers who plan the whole construction keeping the safety and requirements of the people. Individuals in architect career in India provides professional services for new constructions, alterations, renovations and several other activities. Individuals in architectural careers in India visit site locations to visualize their projects and prepare scaled drawings to submit to a client or employer as a design. Individuals in architecture careers also estimate build costs, materials needed, and the projected time frame to complete a build.

2 Jobs Available
##### Landscape Architect

Having a landscape architecture career, you are involved in site analysis, site inventory, land planning, planting design, grading, stormwater management, suitable design, and construction specification. Frederick Law Olmsted, the designer of Central Park in New York introduced the title “landscape architect”. The Australian Institute of Landscape Architects (AILA) proclaims that "Landscape Architects research, plan, design and advise on the stewardship, conservation and sustainability of development of the environment and spaces, both within and beyond the built environment". Therefore, individuals who opt for a career as a landscape architect are those who are educated and experienced in landscape architecture. Students need to pursue various landscape architecture degrees, such as M.Des, M.Plan to become landscape architects. If you have more questions regarding a career as a landscape architect or how to become a landscape architect then you can read the article to get your doubts cleared.

2 Jobs Available
##### Plumber

An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.

2 Jobs Available
##### Orthotist and Prosthetist

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

6 Jobs Available
##### Veterinary Doctor

A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.

5 Jobs Available
##### Pathologist

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

5 Jobs Available
##### Gynaecologist

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.

4 Jobs Available
##### Surgical Technologist

When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.

3 Jobs Available
##### Oncologist

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

3 Jobs Available
##### Chemical Pathologist

Are you searching for a chemical pathologist job description? A chemical pathologist is a skilled professional in healthcare who utilises biochemical laboratory tests to diagnose disease by analysing the levels of various components or constituents in the patient’s body fluid.

2 Jobs Available
##### Biochemical Engineer

A Biochemical Engineer is a professional involved in the study of proteins, viruses, cells and other biological substances. He or she utilises his or her scientific knowledge to develop products, medicines or ways to improve quality and refine processes. A Biochemical Engineer studies chemical functions occurring in a living organism’s body. He or she utilises the observed knowledge to alter the composition of products and develop new processes. A Biochemical Engineer may develop biofuels or environmentally friendly methods to dispose of waste generated by industries.

2 Jobs Available
##### Actor

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.

4 Jobs Available
##### Acrobat

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available
##### Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages. Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available
##### Talent Agent

The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

3 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
##### Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
##### Choreographer

The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

2 Jobs Available
##### Talent Director

Individuals who opt for a career as a talent director are professionals who work in the entertainment industry. He or she is responsible for finding out the right talent through auditions for films, theatre productions, or shows. A talented director possesses strong knowledge of computer software used in filmmaking, CGI and animation. A talent acquisition director keeps himself or herself updated on various technical aspects such as lighting, camera angles and shots.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
##### Content Writer

Content writing is meant to speak directly with a particular audience, such as customers, potential customers, investors, employees, or other stakeholders. The main aim of professional content writers is to speak to their targeted audience and if it is not then it is not doing its job. There are numerous kinds of the content present on the website and each is different based on the service or the product it is used for.

2 Jobs Available
##### Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

2 Jobs Available
##### Linguist

Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian

2 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Engineer

A career as Production Engineer is crucial in the manufacturing industry. He or she ensures the functionality of production equipment and machinery to improve productivity and minimize production costs in order to drive revenues and increase profitability.

2 Jobs Available
##### Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.

2 Jobs Available
##### Product Designer

Individuals who opt for a career as product designers are responsible for designing the components and overall product concerning its shape, size, and material used in manufacturing. They are responsible for the aesthetic appearance of the product. A product designer uses his or her creative skills to give a product its final outlook and ensures the functionality of the design.

Students can opt for various product design degrees such as B.Des and M.Des to become product designers. Industrial product designer prepares 3D models of designs for approval and discusses them with clients and other colleagues. Individuals who opt for a career as a product designer estimate the total cost involved in designing.

2 Jobs Available
##### R&D Personnel

A career as R&D Personnel requires researching, planning, and implementing new programs and protocols into their organization and overseeing new products’ development. He or she uses his or her creative abilities to improve the existing products as per the requirements of the target market.

2 Jobs Available
##### Commercial Manager

A Commercial Manager negotiates, advises and secures information about pricing for commercial contracts. He or she is responsible for developing financial plans in order to maximise the business's profitability.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Computer System Analyst

Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

2 Jobs Available
##### Test Manager

A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures.

2 Jobs Available
##### Azure Developer

A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service.

2 Jobs Available
##### Deep Learning Engineer

A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources.  A Career as a Deep Learning Engineer needs to help the  data scientists and analysts to create effective data sets.

2 Jobs Available