Question

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

• A. $|\mathbf{a}|^{2}$
• B. $2|\mathbf{a}|^{2}$
• C. $3|\mathbf{a}|^{2}$
• D. $4|\mathbf{a}|^{2}$

Answer 1

Answer: (B)

$2|\mathbf{a}|^{2}$

Answer 2

$|\mathbf{a} \times \mathbf{i}|^{2} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_{1} & a_{2} & a_{3} \\ 1 & 0 & 0 \end{matrix} \right|^{2}$, $(\text{Since}\mathbf{a} = a_{1}\mathbf{i} + a_{2}\mathbf{j} + a_{3}\mathbf{k})$

$= |a_{3}\mathbf{j} - a_{2}\mathbf{k}|^{2} = a_{3}^{2} + a_{2}^{2}$

Similarly, $|\mathbf{a} \times \mathbf{j}|^{2} = a_{1}^{2} + a_{3}^{2}$ and $|\mathbf{a} \times \mathbf{k}|^{2} = a_{1}^{2} + a_{2}^{2}$

Hence the required result can be given as

$2(a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) = 2|\mathbf{a}|^{2}.$