Question

The angle between the planes $x+y+z=0$ and $3x-4y+5z=0$ is

• A. ${\cos }^{-1}\left(\frac{1}{5}\sqrt{\frac{2}{5}}\right)$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. ${\cos }^{-1}\left(\frac{2}{5}\sqrt{\frac{2}{3}}\right)$

Answer 1

The correct option is D ${\cos }^{-1}\left(\frac{2}{5}\sqrt{\frac{2}{3}}\right)$

The angle between $x+y+z=0$ & $3x-4y+5z=0$ is
$\theta ={\cos }^{-1}\left[\frac{(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}).(3\overrightarrow{i}-4\overrightarrow{j}+5\overrightarrow{k})}{\sqrt{(1^2+1^2+1^2)(3^2+4^2+5^2)}}\right]$

$={\cos }^{-1}\left(\frac{2}{5}\sqrt{\frac{2}{3}}\right)$
Ans: D