Question

The angle between the planes $2x+y-2z+3=0$ and $6x+3y+2z=5$ is

• A. ${\cos }^{-1}\left(\frac{7}{11}\right)$
• B. ${\cos }^{-1}\left(\frac{9}{21}\right)$
• C. ${\cos }^{-1}\left(\frac{11}{21}\right)$
• D. ${\cos }^{-1}\left(\frac{13}{29}\right)$

Answer 1

The correct option is C ${\cos }^{-1}\left(\frac{11}{21}\right)$

Given planes, $P_1:2x+y-2z+3=0$ and $P_2:6x+3y+2z=5$
$D.r^{\prime }s$ of normal to $P_1:(2,1,-2)$ and of $P_2:(6,3,2)$
So, $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot \sqrt{a_2^2+b_2^2+c_2^2}}$
$\Rightarrow \cos \theta =\frac{11}{21}\Rightarrow \theta ={\cos }^{-1}\left(\frac{11}{21}\right)$