A) \[|\overrightarrow{a}{{|}^{2}}\]
B) \[2|\overrightarrow{a}{{|}^{2}}\]
C) \[3|\overrightarrow{a}{{|}^{2}}\]
D) None of these
Correct Answer: B
Solution :
Let \[\overrightarrow{a}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k}\] Then, \[\overrightarrow{a}\times \hat{i}=({{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k})\times \hat{i}\] \[\Rightarrow \] \[\overrightarrow{a}\times \hat{i}=-{{a}_{2}}\hat{k}+{{a}_{3}}\hat{j}\] ?(i) and \[\overrightarrow{a}\times \hat{j}=({{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k})\times \hat{j}\] \[\Rightarrow \] \[\overrightarrow{a}\times \hat{j}={{a}_{1}}\hat{k}-{{a}_{3}}\hat{i}\] ?(ii) \[\overrightarrow{a}\times \hat{k}=({{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k})\times \hat{k}\] \[\Rightarrow \] \[\overrightarrow{a}\times \hat{k}=-{{a}_{1}}\hat{j}+{{a}_{2}}\hat{i}\] ...(iii) Now, \[|\overrightarrow{a}\times \hat{i}{{|}^{2}}+|\overrightarrow{a}\times \hat{j}{{|}^{2}}+|\overrightarrow{a}\times \hat{k}{{|}^{2}}\] \[=(a_{2}^{2}+a_{3}^{2})+(a_{1}^{2}+a_{3}^{2})+(a_{1}^{2}+a_{2}^{2})\] [using Eqs. (i), (ii) and (iii)] \[=2(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})\] \[=2|\overrightarrow{a}{{|}^{2}}\]
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