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, ${\left|\vec{p}\times \hat{j}\right|}^2=p_3^2+p_1^2,{\left|\vec{p}\times \hat{k}\right|}^2=p_1^2+p_2^2$
$\therefore \frac{3}{2}\left\{{\left|\vec{p}\times \hat{i}\right|}^2+{\left|\vec{p}\times \hat{j}\right|}^2+{\left|\vec{p}\times \hat{k}\right|}^2\right\}$
$=3\left(p_1^2+p_2^2+p_3^2\right)=3{\vec{p}}^2$