Question

A plane $\pi_1$ contains the vectors $\vec{i}+\vec{j}$ and $\vec{i}+2\vec{j}$. Another plane $\pi_2$ contains the vectors $2\vec{i}-\vec{j}$ and $3\vec{i}+2\vec{k}$. $\vec{a}$ is a vector parallel to the line of intersection of $\pi_1$ and $\pi_2$. If the angle $\theta$ between $\vec{a}$ and $\vec{i}-2\vec{j}+2\vec{k}$ is acute, then $\theta=$

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{4}$
• C. $\cos^{-1}(\frac{4}{3\sqrt{5}})$
• D. $\cos^{-1}(\frac{2}{\sqrt{5}})$

Hint:

The direction vector of the line of intersection of two planes is always perpendicular to both of their normal vectors. Therefore, it can be found by taking the cross product of the normal vectors: $\vec{d} = \vec{n_1} \times \vec{n_2}$.

Answer 1

The Correct Option is C