Question

If $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0},$ then which relation is correct

• A. $\mathbf{a} = \mathbf{b} = \mathbf{c} = \mathbf{0}$
• B. $\mathbf{a}.\mathbf{b} = \mathbf{b}.\mathbf{c} = \mathbf{c}.\mathbf{a}$
• C. $\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}$
• D. None of these

Answer 1

Answer: (C)

$\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}$

Answer 2

Since $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$

$\Rightarrow \mathbf{a} \times (\mathbf{a} + \mathbf{b} + \mathbf{c}) = \mathbf{0} \Rightarrow \mathbf{a} \times \mathbf{a} + \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} = \mathbf{0}$

$\Rightarrow \mathbf{a} \times \mathbf{b} = - \mathbf{a} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}$ .....(i)

Similarly, $\mathbf{b} \times (\mathbf{a} + \mathbf{b} + \mathbf{c}) = \mathbf{0} \Rightarrow \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c}$ .....(ii)

By (i) and (ii), we get $\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}.$