Question

For any vector $\vec{p}, $ the value of
$\frac{3}{2}\left\{\left|\vec{p}\times\hat{i}\right|^{2} +\left|\vec{p}\times\hat{j}\right|^{2}+\left|\vec{p}\times\hat{k}\right|^{2}\right\} is$

• A. $\vec{p}^{2}$
• B. $2 \vec{p}^{2}$
• C. $3 \vec{p}^{2}$
• D. $4 \vec{p}^{2}$

Answer 1

Answer: (C)

Suppose $\vec{p}=p_{1}\hat{i}+p_{2} \hat{j}+p_{3} \hat{k}$
$\vec{p}\times\hat{i}=p_{2} \hat{j}\times\hat{i}+p_{3}\hat{k}\times\hat{i}=-p_{2}\hat{k}+p_{3}\hat{j}$
$\left|\vec{p}\times\hat{i}\right|^{2}=p_{2}^{2}+p_{3}^{2}$
Similarly, $\left|\vec{p}\times\hat{j}\right|^{2}=p_{3}^{2}+p_{1}^{2}, \left|\vec{p}\times\hat{k}\right|^{2}=p_{1}^{2}+p_{2}^{2}$
$\therefore \, \frac{3}{2}\left\{\left|\vec{p}\times\hat{i}\right|^{2}+\left|\vec{p}\times\hat{j}\right|^{2}+\left|\vec{p}\times\hat{k}\right|^{2}\right\}$
$=3\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)=3\vec{p}^{2}$