Q119 / 180 · COMEDK 2012 · Mathematics

If $|\mathbf{a}|=2,|\mathbf{b}|=7$ and $\mathbf{a} \times \mathbf{b}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$, then the angle between and $b$ is

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

Solution

We have, $|a|=2,|b|=7$ and $x=3 \hat{\mathfrak{i}}+2 \hat{\hat{j}}+6 \hat{\mathrm{M}}$ So, $|a \times b|=\sqrt{9+4+36}=\sqrt{49}=7$ Since, $\mid$ a $\times \boldsymbol{b}|=|$ a || $\mathrm{D}|\sin \theta| \hat{\mathbf{n}} \mid$ $\Rightarrow \quad 7=(2)(7) \sin \theta \cdot 1[\because|\hat{\mathbf{n}}|=1]$ $\Rightarrow \quad \sin \theta=\frac{1}{2} \Rightarrow \sin \theta=\sin \frac{\pi}{6} \Rightarrow \quad \theta=\frac{\pi}{6}$

Hence, the correct answer is option: 'B'.

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