RD Sharma Solutions Class 12 Mathematics Chapter 26 VSA

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RD Sharma Solutions Class 12 Mathematics Chapter 26 VSA

RD Sharma Solutions Class 12 Mathematics Chapter 26 VSA

Satyajeet KumarUpdated on 24 Jan 2022, 07:35 PM IST

RD Sharma books are widely considered the best material for maths as they are comprehensive and cover all essential concepts. Class 12 RD Sharma chapter 26 exercise VSA solution contains the best answers for the RD Sharma book. Students can take advantage of this material as it is prepared by experts who have years of experience with CBSE exam paper patterns.

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RD Sharma Class 12 Solutions Chapter26 VSA Directions Cosines and Directions Ratios - Other Exercise
Direction Cosines and Direction Ratios Excercise: VSA
RD Sharma Chapter wise Solutions

Students can use this material as a guide and come back to it whenever they don't understand a question. This is an efficient alternative to textbooks as it contains all questions and answers in the same place. RD Sharma Solutions However, as maths includes hundreds of questions, it gets confusing for students to remember every concept. To help them overcome this issue, Careers360 has prepared this material.

RD Sharma Class 12 Solutions Chapter26 VSA Directions Cosines and Directions Ratios - Other Exercise

Direction Cosines and Direction Ratios Excercise: VSA

Directions Cosines and Direction Ratios Exercise Very Short Answer Question question 1

Answer:

We need to define it.
Hint:
Cosines are trigonometric functions.
Given:
Direction cosines of a directed line.
Solution:
The directed cosine of a directed line are the cosines of the angles which the lines makes with the positive direction of the co-ordinate axis.
$\begin{aligned} &l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}} \\ &m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}} \\ &n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} \end{aligned}$

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 2

Answer:
(1,0,0)
Hint:
Cosines of X-axis.
Given:
Cosines of x-axis.
Solution:
The x-axis makes the angle $0^{\circ}, 90^{\circ} \: and\; 90^{\circ}$ with x, y and z-axis
∴ direction cosine of x-axis
$= \cos 0^{\circ}, \cos 90^{\circ} \; and\; \cos 90^{\circ}$
=1, 0,0

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 3.

Answer:
(0, 1,0)
Hint:
Cosines are trigonometric function.
Given:
Cosines of y-axis.
Solution:
The y-axis makes the angle $90^{\circ}, 0^{\circ} and \; 90^{\circ}$ with x, y and z-axis
∴ direction cosine of y-axis
$= \cos 90^{\circ}, \cos 0^{\circ} and \; \cos 90^{\circ}$
=0, 1,0

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 4.

Answer:
(0,0,1)
Hint:
Cosines of z-axis.
Given:
Cosines of z-axis.
Solution:
The z-axis makes the angle $90^{\circ}, 90^{\circ} and \; 0^{\circ}$ with x, y and z-axis
∴direction cosine of z-axis
$= \cos 90^{\circ}, \cos 90^{\circ} and \; \cos 0^{\circ}$
=0, 0,1

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 5.

Answer:
7, -2, 3
Hint:
Distance $=\left|\frac{a x_{1}+b y_{1}+c z_{1}+d}{\sqrt{a^{2}+b^{2}+c^{2}}}\right|$
Given:
P(7, -2, 3) from xy plane
Solution:
Distance of P(7, -2, 3) from xy-plane
i.e, $z=0, a=0, b=0, c=1, d=0$
Distance$=\left|\frac{0 \times 7+0 \times-2+1 \times 3+0}{\sqrt{0^{2}+0^{2}+1^{2}}}\right|=3$
Distance of P(7, -2, 3) from yz-plane
i.e, x=0, a=1, b=0, c=0, d=0
Distance$=\left|\frac{1 \times 7+0 \times-2+0 \times 3}{\sqrt{1^{2}+0^{2}+0^{2}}}\right|=7$
Distance of P(7, -2, 3) from y, xz plane is -2

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 6.

Answer:
13 units
Hint:
$d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}}$
Given:
$P(3, -5, 12 )$
Solution:
The direction cosines of x-axis is $b_1 (1, 0, 0)$
Let $A(\lambda, 0,0)$ be the point on x-axis.
From $P(3, -5, 12 )$ drawn a line on x-axis perpendicularly in x-axis at $A(\lambda, 0,0)$.
Distance of cosines of $AP=b_2\left ( 3-\lambda ,-5, 12 \right )$
we know that
$\begin{aligned} &\overrightarrow{b_{1}} \cdot \overrightarrow{b_{2}}=0 \\ &\Rightarrow 1 \times(3-\lambda)=0 \\ &\Rightarrow 3-\lambda=0 \\ &\Rightarrow \lambda=3 \\ &\therefore A=(3,0,0) \\ &A P=\sqrt{(3-3)^{2}+(0+5)^{2}+(12-0)^{2}} \\ &=\sqrt{25+144} \\ &=\sqrt{169} \\ &=13 \end{aligned}$

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 7

Answer:

2:3
Hint:
$x=\frac{m x_{2}+n x_{1}}{m+n}$
Given:
line segment P(-2, 5, 9) and Q(3, -2, 9)
Solution:
Let R(0, y, z) be the point on y-z plane which divide PQ.
Let the ratio be m:n.
$\begin{aligned} &x=\frac{m x_{2}+n x_{1}}{m+n} \\ &y=\frac{m y_{2}-n y_{1}}{m+n} \\ &z=\frac{m z_{2}-n z_{1}}{m+n} \\ &0=\frac{3 m-2 n}{m+n} \\ &0=3 m-2 n \\ &3 m=2 n \\ &\Rightarrow \frac{m}{n}=\frac{2}{3} \\ &\Rightarrow m: n=2: 3 \end{aligned}$

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 8.

Answer:
$45^{\circ}$
Hint:
$l^{2}+m^{2}+n^{2}=1$
Given:
Line makes $60^{\circ}$ with x-axis and y-axis.
Solution:
$\begin{array}{ll} l=\cos \alpha, & \alpha \rightarrow x-\text { axis } \\ \mathrm{m}=\cos \beta, & \beta \rightarrow y-\text { axis } \\ \mathrm{n}=\cos \gamma, & \gamma \rightarrow z-\text { axis } \end{array}$
We know that
$\begin{aligned} &l^{2}+m^{2}+n^{2}=1 \\ &\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\ &\Rightarrow \cos ^{2} 60^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} \gamma=1\left(\alpha=\beta=60^{\circ}\right) \\ &\Rightarrow \cos ^{2} \gamma=1-2 \cos ^{2} 60^{\circ} \\ &=1-2\left(\frac{1}{4}\right) \end{aligned}$
$\begin{aligned} &=1-\frac{1}{2} \\ &=\frac{1}{2} \\ &\Rightarrow \cos \gamma=\frac{1}{\sqrt{2}} \\ &\gamma=45^{\circ} \end{aligned}$

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 9.

Answer:
-1
Hint:
$\cos ^2\alpha +\cos ^2\beta +\cos^ 2\gamma=1$
Given:
The line makes angle $\alpha ,\beta ,\gamma$ with co-ordinate axis.
Solution:
$\begin{aligned} &=\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma \\ &=2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1 \\ &=2\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right) \\ &=2(1)-3 \\ &=-1 \end{aligned}$

Directions Cosines and Direction Ratios Exercise Very Short Answer Question, question 10.

Answer:
$c:b$
Hint:
$z=\frac{mz_2+nz_1}{m+n}$
Given:
Two points (a, b, c) and (-a,- c, -b)
Solution:
Let the plane xy divided P(a, b, c) and Q (-a,- c, -b) in (x, y, 0)
Let m:n be the ratio.
$\begin{aligned} &x=\frac{m x_{2}+n x_{1}}{m+n} \\ &y=\frac{m y_{2}+n y_{1}}{m+n} \\ &z=\frac{m z_{2}+n z_{1}}{m+n} \\ &0=\frac{m(-b)+n(c)}{m+n} \\ &0=-b m+c n \\ &b m=c n \\ &\Rightarrow \frac{m}{n}=\frac{c}{b} \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 11

Answer:
$\frac{3\pi}{4}$
Hint:
Cosine is trigonometric ratio
Given:
Distance ratio (0, -1, -1)
Solution:
here, a=0, b=1, c= -1
let direction cosine are l, m, n
$\begin{aligned} &l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}} \\ &m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}} \\ &n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} \end{aligned}$
$\begin{aligned} &\cos \theta=n=\frac{(-1)}{\sqrt{(0)^{2}+(1)^{2}+(-1)^{2}}} \\ &\cos \theta=-\frac{1}{\sqrt{2}} \\ &\theta=\frac{3 \pi}{4} \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 12

Answer:
$90^{\circ}$
Hint:
$\cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \cdot \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$
Given:
Distance ratio are proportional to (1, -2, 1) and(4, 3, 2).
Solution:
Here, $a_1, b_1, c_1= 1, -2, 1$
$a_2, b_2, c_2= 4, 3, 2$
We know that,
$\begin{aligned} &\cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \cdot \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} \\ &\therefore a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2} \\ &=(1)(4)+(-2)(3)+(1)(2) \\ &=4-6+2 \\ &=0 \\ &\cos \theta=0 \\ &\theta=90^{\circ} \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 13

Answer:
z units
Hint:
z=0
Given:
P(x, y, z)
Solution:
In xy plane z co-ordinate is zero.
By distance formula.
$\begin{aligned} &d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}} \\ &=\sqrt{(x-x)^{2}+(y-y)^{2}+(z-0)^{2}} \\ &=\sqrt{(z)^{2}} \\ &=z \text { units } \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 14

Answer:
x, 0, z
Hint:
y=0
Given:
Px, y, z on XOZ plane
Solution:
Projection of P on XZ plane
x component = x
y component = 0
z component = z
∴ co-ordinate=x, 0, z

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 15

Answer:
(0, -3, 0)
Hint:
x=0, z=0
Given:
P(2,-3, 5) on y-axis
Solution:
Projection of P on y-axis
x component = 0
z component = 3
y component = 0
∴ co-ordinate=0, -3, 0

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 16

Answer:
5 units
Hint:
$d=\sqrt{y^2+z^2 }$
Given:
P(2, 3, 4) from x-axis.
Solution:
A general point (x, y, z) is at a distance of $\sqrt{y^2+z^2 }$ from x-axis.
Distance of the point (2, 3, 4) from x-axis.
$=\sqrt{3^2+4^2}$
$=25$
=5 units

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 17

Answer:
$\frac{2}{3},\frac{-1}{3},\frac{-2}{3}$
Hint:
$l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}$

Given:
D'R proportional to 2, -1, 2.
Solution:
Let l, m, n be direction cosine.
$\begin{aligned} &l=\frac{2}{\sqrt{4+1+4}} \\ &m=\frac{-1}{\sqrt{4+1+4}} \\ &n=\frac{-2}{\sqrt{4+1+4}} \\ &l=\frac{2}{3} \\ &m=\frac{-1}{3} \\ &n=\frac{-2}{3} \\ &\langle l, m, n\rangle=\langle \frac{2}{3},\frac{-1}{3},\frac{-2}{3}\rangle \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 18

Answer:

0, 0, 1
Hint:
Direction cosines of z-axis is 0,0,1.
Given:
Line parallel z-axis
Solution:
Any line parallel z-axis is
$\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-0}{1}$
$\therefore$ Direction cosines 0,0,1

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 19

Answer:
$\frac{\pi}{3}$
Hint:
$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
Given:
a makes angle $\frac{\pi}{3}$ with i,$\frac{\pi}{4}$ with j and $\theta$ with k
Solution:
Angle of a with $i=\frac{\pi}{3}$
$\begin{aligned} &\Rightarrow(x \hat{\imath}+y \hat{\jmath}+z \hat{k}) \cdot(1 \hat{\imath}+0 \hat{\jmath}+0 \hat{k})=1 \times 1 \times \frac{1}{2}, \; \; \; \; \; \; \; \; \; \; \quad \text { since }|a|=1 \\ &\Rightarrow x \times 1=\frac{1}{2} \\ &\Rightarrow x=\frac{1}{2} \end{aligned}$
Similarly,
$\gamma =\frac{1}{\sqrt{2}}$
let $\gamma$ be the angle.
$\begin{aligned} &\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\ &\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{\sqrt{2}}\right)^{2}+\cos ^{2} \gamma=1 \\ &\cos ^{2} \gamma=\frac{1}{4} \\ &\Rightarrow \cos \gamma=\pm \frac{1}{2} \\ &\gamma=\frac{\pi}{3} \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 20

Answer:
$\sqrt{b^2+c^2}$
Hint:
Distance$=\sqrt{y^2+z^2}$
Given:
P(a,b,c) from x-axis.
Solution:
A general point (x, y, z) is at a distance of $\sqrt{y^2+z^2}$ from x-axis.
Distance of the point (a,b,c) from x-axis $\sqrt{b^2+c^2}$.

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 21

Answer:
$30^{\circ}$
Hint:
$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
Given:
Lines makes $90^{\circ}$ and $60^{\circ}$ with x-axis and y-axis.
Solution:
Here
$\alpha =90^{\circ}$
$\beta =60^{\circ}$
We know that
$\begin{aligned} &\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\ &\Rightarrow \cos ^{2} 90^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} \gamma=1 \\ &\Rightarrow 0+\left(\frac{1}{2}\right)^{2}+\cos ^{2} \gamma=1 \\ &\Rightarrow \cos ^{2} \gamma=1-\frac{1}{4} \\ &=\frac{3}{4} \\ &\cos \gamma=\frac{\sqrt{3}}{2} \mid \\ &\gamma=30^{\circ} \end{aligned}$

Direction Cosines and Direction Ratios Exercise Very Short Answer Question, question 22

Answer:
$\left(0,-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
Hint:
Cosines are trigonometric ratio.
Given:
Lines makes $90^{\circ},135^{\circ}$ and $45^{\circ}$ with x, y and z-axis.
Solution:
Direction cosines $=\cos 90^{\circ},\cos 135^{\circ},\cos 45^{\circ}$
$=\left(0,-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$

RD Sharma class 12th exercise VSA provides students with easy-to-understand answers that are exam-oriented. RD Sharma class 12 solutions chapter 26 exercise VSA follows the CBSE syllabus and is updated to the latest version. There are a total of 22 questions given in the RD Sharma class 12th exercise VSA. The concepts used in this exercise are direction ratios, direction cosines, the distance of a point from the x-axis, the acute angle made by the line with the Z-axis, and many more.

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