Question

If a and b are two vectors, then $(\mathbf{a} \times \mathbf{b})^{2}$ equals

• A. $\left| \begin{matrix} \mathbf{a}.\mathbf{b} & \mathbf{a}.\mathbf{a} \\ \mathbf{b}.\mathbf{b} & \mathbf{b}.\mathbf{a} \end{matrix} \right|$
• B. $\left| \begin{matrix} \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{b} \\ \mathbf{b}.\mathbf{a} & \mathbf{b}.\mathbf{b} \end{matrix} \right|$
• C. $\left| \begin{matrix} \mathbf{a}.\mathbf{b} \\ \mathbf{b}.\mathbf{a} \end{matrix} \right|$
• D. None of these

Answer 1

Answer: (B)

$\left| \begin{matrix} \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{b} \\ \mathbf{b}.\mathbf{a} & \mathbf{b}.\mathbf{b} \end{matrix} \right|$

Answer 2

$(\mathbf{a} \times \mathbf{b})^{2} = a^{2}b^{2} - (\mathbf{a}.\mathbf{b})^{2} = \left| \begin{matrix} \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{b} \\ \mathbf{a}.\mathbf{b} & \mathbf{b}.\mathbf{b} \end{matrix} \right|.$