RD Sharma Solutions Class 12 Mathematics Chapter 26 VSA

RD Sharma Solutions Class 12 Mathematics Chapter 26 VSA

Edited By Satyajeet Kumar | Updated on Jan 24, 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.

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.

Direction Cosines and Direction Ratios Excercise: VSA

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

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

(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.

(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.

(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.

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.

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

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.

$45^{\circ}$
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.

-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.

$c:b$
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

$\frac{3\pi}{4}$
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

$90^{\circ}$
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

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

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

(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

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

$\frac{2}{3},\frac{-1}{3},\frac{-2}{3}$
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

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

$\frac{\pi}{3}$
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

$\sqrt{b^2+c^2}$
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

$30^{\circ}$
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

$\left(0,-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
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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The total number of questions present in chapter 26 are 22. You can find the right solutions for these questions in the RD Sharma class 12 chapter 26 ex VSA solution book.

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