Find the angle between the plane: x + y - 2z = 3 and 2x - 2y + z … - Vidyadip

Question

Find the angle between the plane:
x + y - 2z = 3 and 2x - 2y + z = 5

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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 - 2z = 3 and 2x - 2y + z = 5 is given by
$\cos\theta=\frac{(1)(2)+(1)(-2)+(-2)(1)}{\sqrt{1^2+1^2+(-2)^2}\sqrt{2^2+(-2)^2+1^2}}$
$=\frac{2-2-2}{\sqrt{1+1+4}\sqrt{4+4+1}}=\frac{-2}{\sqrt{6}\sqrt{9}}$
$=\frac{-2}{\sqrt{6}\sqrt{9}}=\frac{-2}{3\sqrt{6}}$
$\theta=\cos^{-1}\Big(\frac{-2}{3\sqrt{6}}\Big)$

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