A) \[|\mathbf{a}{{|}^{2}}\]
B) \[2|\mathbf{a}{{|}^{2}}\]
C) \[3|\mathbf{a}{{|}^{2}}\]
D) None of these
Correct Answer: B
Solution :
Let\[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k}\], then \[\mathbf{a}\times \mathbf{i}=-{{a}_{2}}\mathbf{k}+{{a}_{3}}\mathbf{j},\,\,\mathbf{a}\times \mathbf{j}={{a}_{1}}\mathbf{k}-{{a}_{3}}\mathbf{i}\] and \[\mathbf{a}\times \mathbf{k}={{a}_{2}}\mathbf{i}-{{a}_{2}}\mathbf{j}\] \[\therefore \] \[|\mathbf{a}\times \mathbf{i}{{|}^{2}}+|\mathbf{a}\times \mathbf{j}{{|}^{2}}+|(\mathbf{a}\times \mathbf{k}){{|}^{2}}\] \[=(a_{2}^{2}+a_{3}^{2})+(a_{1}^{2}+a_{3}^{2})+(a_{1}^{2}+a_{2}^{2})\] \[=2(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})=2|\mathbf{a}{{|}^{2}}\]
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