Question

A plane$P_1$ has the equation $2x-y+z=4$ and the plane $P_2$ has the equation $x+ny+2z=11.$ If the angle between $P_1$ and $P_2$ is $\pi /3$ then the value (s) of '$n$' is (are)

• A. $7/2$
• B. $17,-1$
• C. $-17,1$
• D. $-7/2$

Answer 1

Answer: (C)

$-17,1$

A plane$P_1$ has the equation $2x-y+z=4$ and the plane $P_2$ has the equation $x+ny+2z=11.$

The direction vector of the normal of the first plane is $2i-j+k$ and second plane is $i+nj+2k$.

The angle between $P_1$ and $P_2$ is $\pi /3$

$\cos \frac{\pi }{3}=\frac{(2i-j+k).(i+nj+2k)}{\sqrt{(2^2+1^2+1^2)(1^2+n^2+2^2)}}$
$\Rightarrow \frac{1}{2}=\frac{4-n}{\sqrt{6(5+n^2)}}$

$\Rightarrow {\cos }^2\left(\frac{\pi }{3}\right)=\frac{(4-n{)}^2}{6(5+n^2)}$

$\Rightarrow n^2+16n-17=0\Rightarrow (n+17)(n-1)=0$

$\Rightarrow n=-17,1$