Q72 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar unit vectors such that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

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

Solution

Since, $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ $$ =(\mathbf{a} \cdot \mathbf{c}) \cdot \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}} \quad (given) $$ So, $\quad \mathbf{a} \cdot \mathbf{b}=\frac{-1}{\sqrt{2}} \Rightarrow \cos \theta=-\frac{1}{\sqrt{2}}$ $\{\because \mathbf{a}$ and $\mathbf{b}$ are unit vector where $\theta$ is angle between vectors $\mathbf{a}$ and $\mathbf{b}\}$ $$ \Rightarrow \quad \theta=\frac{3 \pi}{4} \text {. } $$

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

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