Q74 / 160 · AP EAMCET 2019 (23 Apr Shift 1) · Mathematics

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three unit vectors such that \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c}\), then the angles between \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{a}, \mathbf{c}\) respectively are

  • A
    \(60^{\circ}, 30^{\circ}\)
  • Correct answer
    B
    \(120^{\circ}, 30^{\circ}\)
  • C
    \(60^{\circ}, 150^{\circ}\)
  • D
    \(120^{\circ}, 150^{\circ}\)

Solution

Given that \(\mathbf{a} \times \mathbf{( b} \times \mathbf{c})=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c}\) \(\Rightarrow \quad \text { (a . c) } \mathbf{b}-\mathbf{( a} \cdot \mathbf{b}) \mathbf{c}=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c}\) Let angle between unit vectors \(\mathbf{a}\) and \(\mathbf{c}\) is ' \(\alpha\) ' and angle between unit vectors \(\mathbf{a}\) and \(\mathbf{b}\) is ' \(\beta\) ', then \(\begin{array}{rlrl} & (\cos \alpha) \mathbf{b}-(\cos \beta) \mathbf{c}=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c} \\ \Rightarrow & \cos \alpha=\frac{\sqrt{3}}{2} \text { and } \cos \beta=-\frac{1}{2} \\ \Rightarrow & & \alpha=30^{\circ} \text { and } \beta=120^{\circ} \end{array}\) Hence, option (b) is correct.

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

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