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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 - 1 = 0 and 2x - y + 3z + 3 = 0
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Answer

The direction ratios of normal to the plane, L1: a1x + b1y + c1z = 0,
are a1, b1, c1 and L2: a2x + b2y + c2z = 0 are a2, b2, c2
$\text{L}_1||\text{L}_2,\ \text{if }\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
$\text{L}_1\perp\text{L}_2,\ \text{if}\ \text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2=0$
The angle between L1 and L2 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 - 1 = 0 and 2x - y + 3z + 3 = 0
Here, a1 = 2, b1 = -1, c1 = 3 and a2 = 2, b2 = -1, c2 = 3
$\frac{\text{a}_1}{\text{a}_2}=\frac{2}{2}=1,\ \frac{\text{b}_1}{\text{b}_2}=\frac{-1}{-1}=1\text{ and } \frac{\text{c}_1}{\text{c}_2}=\frac{3}{3}=1$
$\therefore\ \ \frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
Thus, the given lines are parallel to each other.

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