Question

If $\mathbf{a} \times \mathbf{b} = \mathbf{c},\mathbf{b} \times \mathbf{c} = \mathbf{a}$ and a, b, c be moduli of the vectors a, b, c respectively, then

• A. $a = 1,b = c$
• B. $c = 1,a = 1$
• C. $b = 2,c = 2a$
• D. $b = 1,c = a$

Answer 1

Answer: (D)

$b = 1,c = a$

Answer 2

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

$\therefore$ a is perpendicular to both b and $\mathbf{c}$ and $\mathbf{c}$ is perpendicular to both $\mathbf{a}$ and b.

∴ $\mathbf{a},\mathbf{b},\mathbf{c}$are mutually perpendicular

Now, $\mathbf{a} = \mathbf{b} \times \mathbf{c} = \mathbf{b} \times (\mathbf{a} \times \mathbf{b}) = (\mathbf{b}.\mathbf{b})\mathbf{a} - (\mathbf{b}.\mathbf{a})\mathbf{b}$

or $\mathbf{a} = b^{2}\mathbf{a} - (\mathbf{b}.\mathbf{a})\mathbf{b} = b^{2}\mathbf{a}$, $\left\{ \because\mathbf{a}\bot\mathbf{b} \right\}$

$\Rightarrow 1 = b^{2}$, $\therefore\mathbf{c} = \mathbf{a} \times \mathbf{b} = ab\sin 90{^\circ}\widehat{\mathbf{n}}$

Take moduli of both sides, then $c = ab$, but $b = 1 \Rightarrow c = a$.