Vectors 4 Question 7

7. Let the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$ and $\overrightarrow{\mathbf{d}}$ be such that $(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{d}})=\overrightarrow{\mathbf{0}}$. If $P _1$ and $P _2$ are planes determined by the pairs of vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$, and $\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{d}}$ respectively, then the angle between $P _1$ and $P _2$ is

(2000, 2M)

(a) 0

(b) $\pi / 4$

(c) $\pi / 3$

(d) $\pi / 2$

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Answer:

Correct Answer: 7. (a)

Solution:

  1. If $\theta$ is the angle between $P _1$ and $P _2$, then normal to the planes are

$ \begin{array}{rl} N _1 & =\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}, \\ N _2 & =\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{d}} \end{array} $

$\therefore \quad N_1 \times N_2= 0$

Then, $| N_1 | \times | N_2 | \sin \theta =0 $

$ \Rightarrow \quad \quad \quad \sin \theta = 0 \Rightarrow \theta =0 $



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