Question

Let $\overset{ \rightarrow }{A}$ be a vector parallel to the line of intersection of the planes $P_{1}$ and $P_{2}$ . The plane $P_{1}$ is parallel to the vectors $2\hat{j}+3\hat{k}$ and $4\hat{j}-3\hat{k}$ while plane $P_{2}$ is parallel to the vectors $\hat{j}-\hat{k}$ and $\hat{i}+\hat{j}$ . The acute angle between $\overset{ \rightarrow }{A}$ and $2\hat{i}+\hat{j}-2\hat{k}$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{5 \pi }{12}$

Answer 1

Answer: (B)

Let, $\overset{ \rightarrow }{a}=2\hat{j}+3\hat{k}$
$\overset{ \rightarrow }{b}=4\hat{j}-3\hat{k}$
$\overset{ \rightarrow }{c}=\hat{j}-\hat{k}$
$\overset{ \rightarrow }{d}=\hat{i}+\hat{j}$
$\overset{ \rightarrow }{e}=2\hat{i}+\hat{j}-2\hat{k}$
A normal vector perpendicular to the plane $P_{1}$ is $\overset{ \rightarrow }{a}\times \overset{ \rightarrow }{b}$
A normal vector perpendicular to the plane $P_{2}$ is $\overset{ \rightarrow }{c}\times \overset{ \rightarrow }{d}$
A vector parallel to the line of intersection is
$\left(\overset{ \rightarrow }{a} \times \overset{ \rightarrow }{b}\right)\times \left(\overset{ \rightarrow }{c} \times \overset{ \rightarrow }{d}\right)=\left(\hat{k} - \hat{j}\right)$
Let, $\theta $ be the acute angle between them then,
$cos\theta =\left|\frac{\overset{ \rightarrow }{A} \cdot \overset{ \rightarrow }{e}}{\left|\overset{ \rightarrow }{A}\right| \left|\overset{ \rightarrow }{e}\right|}\right|=\left|\frac{\left(- 3\right)}{\sqrt{2} \times 3}\right|=\frac{1}{\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{4}$