Question

If a, b, c are three vectors such that $\mathbf{a} = \mathbf{b} + \mathbf{c}$ and the angle between b and c is $\pi/2,$ then

• A. $a^{2} = b^{2} + c^{2}$
• B. $b^{2} = c^{2} + a^{2}$
• C. $c^{2} = a^{2} + b^{2}$
• D. $2a^{2} - b^{2} = c^{2}$ (Note : Here $a = |\mathbf{a}|,b = |\mathbf{b}|,c = |\mathbf{c}|)$

Answer 1

Answer: (A)

$a^{2} = b^{2} + c^{2}$

Answer 2

Given that $\Rightarrow \mathbf{a} \times \mathbf{b} = \mathbf{c}$ and angle between b and c is $\frac{\pi}{2}$.

So, $\mathbf{a}^{2} = \mathbf{b}^{2} + \mathbf{c}^{2} + 2\mathbf{b}.\mathbf{c}$

or $\mathbf{a}^{\mathbf{2}}\mathbf{=}\mathbf{b}^{\mathbf{2}}\mathbf{+}\mathbf{c}^{\mathbf{2}}\mathbf{+ 2|b||c|}\mathbf{\cos}\frac{\mathbf{\pi}}{\mathbf{2}}$

or $\mathbf{a}^{\mathbf{2}}\mathbf{=}\mathbf{b}^{\mathbf{2}}\mathbf{+}\mathbf{c}^{\mathbf{2}}\mathbf{+ 0,}\mathbf{\therefore}\mathbf{a}^{\mathbf{2}}\mathbf{=}\mathbf{b}^{\mathbf{2}}\mathbf{+}\mathbf{c}^{\mathbf{2}}$

i.e., $a^{2} = b^{2} + c^{2}$.