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

$\cos \theta=\frac{(\hat{i}+\hat{j}+2 \hat{k}) \cdot(2 \hat{i}-\hat{j}+\hat{k})}{6}=\frac{2-1+2}{6}=\frac{1}{2}$

$\theta=\pi / 3$

$\alpha=\pi / 6$

$(\tan ^{2} \theta)(\cot ^{2} \alpha) = 9$