Question

The angle between the planes 2x - y + z = 6 and x + y + 2z = 7 is:

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (C)

Note: Angle between the planes
$a_{1}x+b_{1}y+c_{1}z+d_1$ and
$a_{2}x+b_{2}y+c_{2}z+d_2$ is given by
$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}}$
Now in the given question:
$\cos \theta =\frac{2\times 1+\left(-1\right)\times 1+1\times 2}{\sqrt{2^2+{\left(-1\right)}^2+1^2}\sqrt{1^2+1^2+2^2}}$
$\Rightarrow \cos \theta =\frac{3}{\sqrt{6}\sqrt{6}}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}.$