Question

If $\mathbf{a}.\mathbf{b} = \mathbf{a}.\mathbf{c},\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}$ and $\mathbf{a} \neq \mathbf{0},$ then

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

Answer 1

Answer: (C)

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

Answer 2

$\mathbf{a}.(\mathbf{b} - \mathbf{c}) = 0,$ $\mathbf{a} \times (\mathbf{b} - \mathbf{c}) = \mathbf{0} \Rightarrow \mathbf{b} = \mathbf{c}$ or $\mathbf{a} = \mathbf{0},$

but $\mathbf{a} \neq \mathbf{0}.$ Hence $\mathbf{b} - \mathbf{c} = \mathbf{0}.$ i.e., $\mathbf{b} = \mathbf{c}$.