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

$ \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)$
$(3) (3)=9$