A) \[{{40}^{o}},\,{{80}^{o}}\]
B) \[{{45}^{o}},\,\,{{45}^{o}}\]
C) \[{{30}^{o}},\,\,{{60}^{o}}\]
D) \[{{90}^{o}},\,\,{{60}^{o}}\]
Correct Answer: D
Solution :
As we know, \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}\] ......(i) \[\because \,\,\,\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}}{2}\] (Given) From equation (i), \[(\mathbf{a}\,.\,\mathbf{c})\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\mathbf{c}=\frac{\mathbf{b}}{2}\] or \[\left( \mathbf{a}.\mathbf{c}-\frac{1}{2} \right)\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}=\mathbf{c}\] Comparison on both sides of b and \[\mathbf{c}\] \[\mathbf{a}\,.\,\mathbf{c}-\frac{1}{2}=0,\] \[\mathbf{a}\,.\,\mathbf{b}=0\] \[\Rightarrow \text{ }|\mathbf{a}||\mathbf{c}|\cos \theta =\frac{1}{2}\Rightarrow (1)(1)\cos \theta =\frac{1}{2}\Rightarrow \theta =60{}^\circ \] or \[\mathbf{a}\,.\,\mathbf{b}=0\], \[\therefore \,\,\,\theta =90{}^\circ \]. So the angle between \[\mathbf{a}\] with b and c are \[90{}^\circ \] and \[60{}^\circ \] respectively.
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