Question

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

• A. 2
• B. 4
• C. 1
• D. 0

Answer 1

The Correct Option is A

We have,
$| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}$
$(| a ||\hat{ i }| \,\sin \,\alpha)^{2}+(| a ||\hat{ j }| \,\sin \,\beta)^{2}+(| a ||\hat{ k }| \,\sin \,\gamma)^{2}$
$=\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \,\gamma \,\,[\because| a |=|\hat{ i }|=|\hat{ j }|=|\hat{ k }|=1]$
$=1-\cos ^{2} \alpha+1-\cos ^{2} \beta+1-\cos ^{2} \gamma$
$=3-\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right)$
$=3-1=2 \left[\because \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\right]$