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If a is a unit vector , then $|a \times \hat{i}|^{2} + |a \times \hat{j}|^{2}+ |a \times \hat{k}|^{2}=$

Answer:

Explanation:

We have,

$|a \times \hat{i}|^{2} + |a \times \hat{j}|^{2}+|a \times \hat{k}|^{2}$

 $(|a||\hat{i}| \sin \alpha)^{2}+(|a||\hat{j}| \sin \beta)^{2}+(|a||\hat{k}| \sin \gamma)^{2}$

$= \sin^{2} \alpha+\sin^{2} \beta+\sin^{2} \gamma$   [ $\because |a|=|\hat{i}|=|\hat{j}|=|\hat{k}|=1]$

$=1- \cos^{2} \alpha+1-\cos^{2} \beta+1-\cos^{2} \gamma$

  $=3-(\cos^{2}  \alpha+\cos^{2} \beta+ \cos ^{2} \gamma)$

= 3-1=2       $[ \because   \cos^{2} \alpha+\cos^{2} \beta+ \cos^{2} \gamma=1]$