Question:
Let \(\vec a,\vec b\) and \(\vec c\) be three unit vectors, out of which vectors \(\vec b\) and \(\vec c\) are non-parallel. If \(\alpha\) and \(\beta\) are the angles which vector \(\vec a\) makes with vectors \(\vec b\) and \(\vec c\) respectively and \(\vec a\times(\vec b\times\vec c)=\frac12\vec b\), then \(|\alpha-\beta|\) is equal to :
Let \(\vec a,\vec b\) and \(\vec c\) be three unit vectors, out of which vectors \(\vec b\) and \(\vec c\) are non-parallel. If \(\alpha\) and \(\beta\) are the angles which vector \(\vec a\) makes with vectors \(\vec b\) and \(\vec c\) respectively and \(\vec a\times(\vec b\times\vec c)=\frac12\vec b\), then \(|\alpha-\beta|\) is equal to :
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Expand \(\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c\) and compare coefficients of the independent vectors \(\vec b,\vec c\).
Updated On: Nov 5, 2026
- \(90^\circ\)
- \(30^\circ\)
- \(45^\circ\)
- \(60^\circ\)
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The Correct Option is B
Solution and Explanation
\(\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c=\frac12\vec b\).
Since \(\vec b,\vec c\) are non-parallel (linearly independent), compare coefficients: \(\vec a\cdot\vec c=\frac12\) and \(\vec a\cdot\vec b=0\).
All are unit vectors: \(\cos\beta=\frac12\Rightarrow\beta=60^\circ\); \(\cos\alpha=0\Rightarrow\alpha=90^\circ\).
\(|\alpha-\beta|=30^\circ\).
Answer: (2) \(30^\circ\)
Since \(\vec b,\vec c\) are non-parallel (linearly independent), compare coefficients: \(\vec a\cdot\vec c=\frac12\) and \(\vec a\cdot\vec b=0\).
All are unit vectors: \(\cos\beta=\frac12\Rightarrow\beta=60^\circ\); \(\cos\alpha=0\Rightarrow\alpha=90^\circ\).
\(|\alpha-\beta|=30^\circ\).
Answer: (2) \(30^\circ\)
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