Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry3 Marks
Question
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
$2x + y + 3z - 2$ and $x - 2y + 5 = 0$
✓

Answer

The direction ratios of normal to the plane,$ L_1: a_1x + b_1y + c_1z = 0,$
are $a_1, b_1, c_1$ and $L_2: a_2x + b_2y + c_2z = 0$ are $a_2, b_2, c_2$​​​​​​​
$\text{L}_1||\text{L}_2,\ \text{if }\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
$\text{L}_1\perp\text{L}_2,\ \text{if}\ \text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2=0$
The angle between $L_1$​​​​​​​ and $L_2$ is given by,
$\text{Q}=\cos^{-1}\Bigg|\frac{\text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2}{\sqrt{\text{a}_1^2+\text{b}_1^2+\text{c}_1^2}.\sqrt{\text{a}_2^2+\text{b}_2^2+\text{c}_2^2}}$
The equations of the planes are $2x + y + 3z - 2 = 0$ and $x - 2y + 5 = 0$
Here, $a_1 = 2, b_1 = 1, c_1 = 3$ and $a_2 = 1, b_2 = -2, c_2 = 0$
$\therefore\ \text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2=2\times1+1\times(-2)+3\times0=0$
Thus, the given planes are perpendicular to each other.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the coordinates of the foot of the perpendicular drawn from the origin.
$3y + 4z - 6 = 0$
→
Evaluate: $\int \frac{(2 x+3)}{\sqrt{x^2+x+1}} d x$
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}'(\text{x})=\text{x}^{3}(2\text{x}-1)^{3}$
→
Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined by $f(x) = x + 1$ and $g(x) = x - 1$. Show that $fog = gof = I_R.$
→
Write the value of $\cos^2\Big(\frac{1}{2}\cos^{-1}\frac{3}{5}\Big).$
→
Differentiate the function $\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0<x<\frac{\pi}{2}$ w.r.t. x.
→
A man wins a rupee for head and loses a rupee for tail when a coin is tossed. Suppose that he tosses once and quits if he wins but tries once more if he loses on the first toss. Find the probability distribution of the number of rupees the man wins.
→
A bag contains $4$ white, $7$ black and $5$ red balls. $4$ balls are drawn with replacement. What is the probability that at least two are white?
→
Evaluate the following integrals:
$\int\frac{\big\{\text{e}^{\sin^{-1}\text{x}}\big\}^2}{\sqrt{1-\text{x}^2}}\text{dx}$
→
Prove that $\tan^{–1} \Bigg(\frac{\text{cos x}}{\text{1 + sinx}}\Bigg)=\frac{\pi}{4}-\frac{x}{2},\text{x}\in\Bigg(-\frac{\pi}{\text{2}},\frac{{\pi}}{2}\Bigg).$
→