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:
- 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 $
