Let $\theta$ be the angle between the planes $P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ and $P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$. Let $\mathrm{L}$ be the line that meets $P_{2}$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_{2}$. If $\alpha$ is the angle between $\mathrm{L}$ and $P_{2}$, then $\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$ is equal to ____________.
Answer (integer)
9
Solution
$P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$
<br/><br/>$$
\begin{aligned}
& P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15 \\\\
& \text { then } \cos \theta=\frac{3}{\sqrt{6} \cdot \sqrt{6}}=\frac{1}{2} \\\\
& \begin{aligned}
& \therefore \theta=\frac{\pi}{3}, \text { Now } \alpha=\frac{\pi}{2}-\theta \\\\
& \therefore \tan ^{2} \theta \cdot \cot ^{2} \alpha=\tan ^{4} \theta \\\\
&=(\sqrt{3})^{4}=9
\end{aligned}
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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