Let the vectors a, b, c and d be such that (a × b) × (c × d) = 0. If P_1 and P_2 are planes determined by the pairs of vectors a, b and c, d respectively, then the angle between P_1 and P_2 is

Let the vectors a, b, c and d be such that (a × b) × (c ×… — JEE Main 2000 Mathematics PYQ

2000

Let the vectors

\vec{a}, \vec{b}, \vec{c}

and

\vec{d}

be such that

(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0}

. If

P_1

and

P_2

are planes determined by the pairs of vectors

\vec{a}, \vec{b}

and

\vec{c}, \vec{d}

respectively, then the angle between

P_1

and

P_2

A.
0
B.
$\pi/4$
C.
$\pi/3$
D.
$\pi/2$

Show answer and solution

Correct option: A

Solution

(a×b) is the normal to plane P₁, and (c×d) is the normal to plane P₂. Their cross product being zero means these two normal vectors are parallel (one is a scalar multiple of the other). Parallel normals imply the planes are parallel (or identical), so the angle between P₁ and P₂ is 0.

[2000]

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