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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    If a, b, 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 a and b is [IIT 1995]

    A) \[\frac{\pi }{4}\]

    B) \[\frac{\pi }{2}\]

    C) \[\frac{3\pi }{4}\]

    D) \[\pi \]

    Correct Answer: C

    Solution :

    • \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\Rightarrow (\mathbf{a}\,.\,\mathbf{c})\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\]                   
    • \[\Rightarrow \left[ (\mathbf{a}\,.\,\mathbf{c})-\frac{1}{\sqrt{2}} \right]\mathbf{b}-\left[ (\mathbf{a}\,.\,\mathbf{b})+\frac{1}{\sqrt{2}} \right]\,\mathbf{c}=0\]
    • \[\Rightarrow \mathbf{a}\,.\,\mathbf{c}=\frac{1}{\sqrt{2}},\] \[\mathbf{a}\,.\,\mathbf{b}=-\frac{1}{\sqrt{2}}\]                   
    • \[\Rightarrow \,|\mathbf{a}|\,|\mathbf{c}|\cos \theta =\frac{1}{\sqrt{2}},\] \[|\mathbf{a}|\,|\mathbf{b}|\cos \varphi =-\frac{1}{\sqrt{2}}\]  
    • \[\Rightarrow \cos \theta =\frac{1}{\sqrt{2}},\] \[\cos \varphi =-\frac{1}{\sqrt{2}}\Rightarrow \theta =\frac{\pi }{4},\] \[\varphi =\frac{3\pi }{4}.\]


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