A) \[|\mathbf{a}{{|}^{2}}\]
B) \[2\,\,|\mathbf{a}{{|}^{2}}\]
C) \[3\,\,|\mathbf{a}{{|}^{2}}\]
D) \[4\,\,|\mathbf{a}{{|}^{2}}\]
Correct Answer: B
Solution :
\[|\mathbf{a}\times \mathbf{i}{{|}^{2}}={{\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ 1 & 0 & 0 \\ \end{matrix} \right|}^{2}}\], \[(\text{Since}\,\,\,\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k})\] \[=\,|{{a}_{3}}\mathbf{j}-{{a}_{2}}\mathbf{k}{{|}^{2}}=a_{3}^{2}+a_{2}^{2}\] Similarly, \[|\mathbf{a}\times \mathbf{j}{{|}^{2}}=a_{1}^{2}+a_{3}^{2}\] and \[|\mathbf{a}\times \mathbf{k}{{|}^{2}}=a_{1}^{2}+a_{2}^{2}\] Hence the required result can be given as \[2(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})=2|\mathbf{a}{{|}^{2}}.\]
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