Question

For any two vectors a and b, $\mathbf{(a}\mathbf{\times}\mathbf{b}\mathbf{)}^{\mathbf{2}}$ is equal to

• A. $a^{2} - b^{2}$
• B. $a^{2} + b^{2}$
• C. $a^{2}b^{2} - (\mathbf{a}.\mathbf{b})^{2}$
• D. None of these

Answer 1

Answer: (C)

$a^{2}b^{2} - (\mathbf{a}.\mathbf{b})^{2}$

Answer 2

$(\mathbf{a} \times \mathbf{b})^{2} = (\mathbf{a} \times \mathbf{b}).(\mathbf{a} \times \mathbf{b}) = (ab\sin\theta\widehat{\mathbf{n}})(ab\sin\theta\widehat{\mathbf{n}})$

$= a^{2}b^{2}\sin^{2}\theta = a^{2}b^{2}(1 - \cos^{2}\theta)$

$= a^{2}b^{2} - a^{2}b^{2}\cos^{2}\theta = a^{2}b^{2} - (\mathbf{a}.\mathbf{b})^{2}.$