A) \[\alpha =\beta \ne \gamma \]
B) \[\alpha =\gamma \ne \beta \]
C) \[\beta =\gamma \ne \alpha \]
D) \[\alpha =\beta =\gamma \]
Correct Answer: D
Solution :
Angle between \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}\] is equal to \[{{\cos }^{-1}}\left\{ \frac{(\mathbf{i}+\mathbf{j}+\mathbf{k})\,.\,\mathbf{i}}{|\mathbf{i}+\mathbf{j}+\mathbf{k}|\,\,|\mathbf{i}|} \right\}\Rightarrow \alpha ={{\cos }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\] Similarly angle between \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{j}\] is \[\beta ={{\cos }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\] and between \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{k}\]is \[\gamma ={{\cos }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\,.\] Hence \[\alpha =\beta =\gamma .\]
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