Three Dimensional Geometry Question 26
Question 26 - 31 January - Shift 1
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 $L$ 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 $L$ and $P_2$ then $(\tan ^{2} \theta)(\cot ^{2} \alpha)$ is equal to __________
Show Answer
Answer: 9
Solution:
Formula: Angle between two planes
$\cos \theta=\frac{(\hat{i}+\hat{j}+2 \hat{k}) \cdot(2 \hat{i}-\hat{j}+\hat{k})}{6}=\frac{2-1+2}{6}=\frac{1}{2}$
$\theta=\pi / 3$
$\alpha=\pi / 6$
$(\tan ^{2} \theta)(\cot ^{2} \alpha) = 9$
