Find the angle between the plane: x - y + z = 5 and x + 2y + z = … - Vidyadip

Question

Find the angle between the plane:
x - y + z = 5 and x + 2y + z = 9

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Answer

We know that the angle between the planes,
$\text{a}_1\text{x}+\text{b}_1\text{y}+\text{c}_1\text{z}+\text{d}_1=0$ and $\text{a}_2\text{x}+\text{b}_2\text{y}+\text{c}_2\text{z}+\text{d}_2=0$ is given by
$\cos\theta=\frac{\text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2}{\sqrt{\text{a}_1^2+\text{b}_1^2+\text{c}_1^2}\sqrt{\text{a}_2^2+\text{b}_2^2+\text{c}^2_2}}$
So, the angle between x - y + z = 5 and x + 2y + z = 9 is given by
$\cos\theta=\frac{(1)(1)+(-1)(2)+(1)(1)}{\sqrt{1^2+(-1)^2+1^2}\sqrt{1^2+2^2+1^2}}$
$=\frac{1-2+1}{\sqrt{1+1+1}\sqrt{1+4+1}}$
$=\frac{0}{\sqrt{3}\sqrt{6}}=0$
$\theta=\cos^{-1}(0)=\frac{\pi}{2}$

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