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
$a_1x + b_1y + c_1z + d_1$ 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:
$\cos\theta = \frac{ 2 \times1+ \left(-1\right)\times 1+1\times 2}{\sqrt{2^{2} + \left(-1\right)^{2} + 1^{2} } \sqrt{1^{2} + 1^{2} + 2^{2}}}$
$ \Rightarrow \cos\theta = \frac{3}{\sqrt{6}\sqrt{6}} = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{3} . $