JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    Let the unit vectors a and b be perpendicular and the unit vector c be inclined at an angle q to both a and b. If \[\mathbf{c}=\alpha \,\mathbf{a}+\beta \,\mathbf{b}+\gamma \,(\mathbf{a}\times \mathbf{b}),\] then [Orissa JEE 2003]

    A) \[\alpha =\beta =\cos \theta ,\,\,{{\gamma }^{2}}=\cos \,\,2\theta \]

    B) \[\alpha =\beta =\cos \theta ,\,\,{{\gamma }^{2}}=-\cos \,\,2\theta \]

    C) \[\alpha =\cos \theta ,\,\,\beta =\sin \theta ,\,\,{{\gamma }^{2}}=\cos \,\,2\theta \]

    D) None of these

    Correct Answer: B

    Solution :

    • \[\mathbf{c}=\alpha \,\mathbf{a}+\beta \,\mathbf{b}+\gamma \,(\mathbf{a}\times \mathbf{b})\]\[\Rightarrow \mathbf{c}\,.\,\mathbf{a}=\alpha \] and \[\mathbf{c}\,.\,\mathbf{b}=\beta \]                   
    • \[\Rightarrow \alpha =\beta =\cos \theta \]                   
    • Also, \[1=\mathbf{c}\,.\,\mathbf{c}\],                                  
    • \[\therefore \left[ \alpha \,\mathbf{a}+\beta \,\mathbf{b}+\gamma \,(\mathbf{a}\times \mathbf{b}) \right]\,.\,\left[ (\alpha \,\mathbf{a}+\beta \,\mathbf{b})+\gamma \,(\mathbf{a}\times \mathbf{b}) \right]=1\]                   
    • \[\Rightarrow 2{{\alpha }^{2}}+{{\gamma }^{2}}{{(\mathbf{a}\times \mathbf{b})}^{2}}=1\],   
    • \[\left\{ \because \,\,\alpha \,=\,\beta  \right\}\]                   
    • \[\Rightarrow 2{{\alpha }^{2}}+{{\gamma }^{2}}[{{\mathbf{a}}^{2}}{{\mathbf{b}}^{2}}-{{(\mathbf{a}.\mathbf{b})}^{2}}]=1\Rightarrow 2{{\alpha }^{2}}+{{\gamma }^{2}}=1\]                   
    • Hence, \[{{\gamma }^{2}}=1-2{{\alpha }^{2}}=1-2\,{{\cos }^{2}}\theta =-\cos 2\theta .\]

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