Question

Let $\theta$ be the angle between the planes $P_1:\vec{r}\cdot (\hat{i}+\hat{j}+2\hat{k})=9$ and $P_2:\hat{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_4$. If $\alpha$ is the angle between $L$ and $P_2$, then $\left({\tan }^2\theta \right)\left({\cot }^2\alpha \right)$ is equal to

Answer 1

Answer: 9

$\cos \theta =\frac{(\hat{i}+\hat{j}+2\hat{k})\cdot (2\hat{i}-\hat{j}+\hat{k})}{6}=\frac{2-1+2}{6}=\frac{1}{2}$
$\theta =\pi /3$
$\alpha =\pi /6$
$\left({\tan }^2\theta \right)\left({\cot }^2\alpha \right)$
$(3)(3)=9$