Question

If a, b, c, d are coplanar vectors, then $(\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) =$

• A. $|\mathbf{a} \times \mathbf{c}|^{2}$
• B. $|\mathbf{a} \times \mathbf{d}|^{2}$
• C. $|\mathbf{b} \times \mathbf{c}|^{2}$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

$(\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) = \lbrack\mathbf{abd}\rbrack\mathbf{c} - \lbrack\mathbf{abc}\rbrack\mathbf{d}$

$\because\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ are coplanar vectors

$\therefore\lbrack\mathbf{abd}\rbrack = \lbrack\mathbf{abc}\rbrack = 0.$ So, $(\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) = \mathbf{0}$.