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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 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.
$7x + 5y + 6z + 30 = 0$ and $3x - y - 10z + 4 = 0$
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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 are $7x + 5y + 6z + 30 = 0$ and
$3x - y - 10z + 4 = 0$
Here, $a_1 = 7, b_1 = 5, c_1 = 6$
$a_2 = 3, b_2 = -1, c_2 = -10$
$a_1a_2 + b_1b_2 + c_1c_2 = 7 \times 3 + 5 \times (-1) + 6 \times (-10)$
$=-44\neq0$
Therefore, the given planes are not perpendicular.
$\frac{\text{a}_1}{\text{a}_2}=\frac{7}{3},\ \frac{\text{b}_1}{\text{b}_2}=\frac{5}{-1}=-5,\ \frac{\text{c}_1}{\text{c}_2}=\frac{6}{-10}=\frac{3}{-5}$
It can be seen that, $\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2},$
Therefore, the given planes are not parallel.
The angle between them is given by,
$\text{Q}=\cos^{-1}\begin{vmatrix}\frac{7\times3+5\times(-1)+6\times(-10)}{\sqrt{(7)^2+(5)^2+(6)^2\times\sqrt{(3)^2+(-1)^2+(-10)^2}}}\end{vmatrix}$
$=\cos^{-1}\Bigg|\frac{21-5-60}{\sqrt{110}\times\sqrt{110}}\Bigg|$
$=\cos^{-1}\frac{44}{110}$
$=\cos^{-1}=\frac{2}{5}.$

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