Vectors 3 Question 17
17. For non-zero vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}|,(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{c}}|$ $=|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}|$ holds, if and only if
(1982, 2M)
(a) $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=0, \overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=0$
(b) $\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=0, \overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}=0$
(c) $\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}=0, \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=0$
(d) $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}=0$
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Answer:
Correct Answer: 17. (d)
Solution:
- Given
$$ \begin{array}{r} \mid \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{c}}|=| \overrightarrow{\mathbf{a}}|| \overrightarrow{\mathbf{b}} | \overrightarrow{\mathbf{c}} \mid \\ \Rightarrow|| \overrightarrow{\mathbf{a}}|| \overrightarrow{\mathbf{b}}|\sin \theta \hat{\mathbf{n}} \cdot \overrightarrow{\mathbf{c}}|=|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}} | \overrightarrow{\mathbf{c}}| \end{array} $$
$\Rightarrow|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}||\sin \theta \cdot \cos \alpha|=|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}|$
$$ \begin{array}{ll} \Rightarrow & |\sin \theta| \cdot|\cos \alpha|=1 \Rightarrow \theta=\frac{\pi}{2} \quad \text { and } \quad \alpha=0 \\ \therefore & \overrightarrow{\mathbf{a}} \perp \overrightarrow{\mathbf{b}} \text { and } \overrightarrow{\mathbf{c}} | \hat{\mathbf{n}} \end{array} $$
i.e. $\overrightarrow{\mathbf{a}} \perp \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ perpendicular to both $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$.
