Let A be vector parallel to line of intersection of planes P_1 and P_2 through origin. P_1 is parallel to the vectors 2 j +3 k and 4 j -3 k and P_2 is parallel to j - k and 3 i +3 j , then the angle between vector A and

Correct answer: B. (π)/(4) Let vector A O be parallel to line of intersection of planes P_1 and P_2 through, i.e. aligned & [(2 j +3 k ) ×(4 j -3 k )] ×[( j - k ) ×(3 i +3 j )]=54( j - k ). & Angle between 54( j - k ) and (2 i + j -2 k ) & θ=±( 54+108 3.54 · 2 )=± 1 2 & θ=(π)/(4), (3 π)/(4) aligned

JEE Advanced 2006

Exam: JEE Advanced
Year: 2006
Subject: Mathematics
Chapter: Vector Algebra

JEE Advanced 2006 · Mathematics · Vector Algebra

Let A be vector parallel to line of intersection of planes P_1 and...

Let

\mathbf{A}

be vector parallel to line of intersection of planes

P_1

and

P_2

through origin.

P_1

is parallel to the vectors

2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}

and

4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}

and

P_2

is parallel to

\hat{\mathbf{j}}-\hat{\mathbf{k}}

and

3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}

, then the angle between vector

\mathbf{A}

and

2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}

Options

A $\frac{\pi}{2}$
B $\frac{\pi}{4}$
C $\frac{\pi}{6}$
D $\frac{3 \pi}{4}$

Correct answer

B. $\frac{\pi}{4}$

Step-by-step solution

Let vector

\mathbf{A O}

be parallel to line of intersection of planes

P_1 and P_2

through, i.e.

\begin{aligned}& {[(2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \times(4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})] \times[(\hat{\mathbf{j}}-\hat{\mathbf{k}}) \times(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}})]=54(\hat{\mathbf{j}}-\hat{\mathbf{k}}).} \\ & \therefore \text{ Angle between }54(\hat{\mathbf{j}}-\hat{\mathbf{k}}) \text{ and }(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \\ & \Rightarrow \cos \theta=\pm\left(\frac{54+108}{3.54 \cdot \sqrt{2}}\right)=\pm \frac{1}{\sqrt{2}}\\ & \therefore \quad \theta=\frac{\pi}{4}, \frac{3 \pi}{4} \end{aligned}

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