Q. If $\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ and three vectors such that $\overrightarrow{\mathbf{a}} = \ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}$ and the angle between $\ \overrightarrow{\mathbf{b}}\ and\ \overrightarrow{\mathbf{c}}$ is $\frac{\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) ${2a}^{2} - b^{2} = c^{2}$
Explanation
(a)
Given, $\overrightarrow{\mathbf{a}} = \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}$
$and\ \overrightarrow{\mathbf{b}}\bot\overrightarrow{\mathbf{c}}$
$then\ \left| \overrightarrow{\mathbf{a}} \right|^{2} = |\overrightarrow{\mathbf{b}}|^{2} + |\overrightarrow{\mathbf{c}}|^{2} + 2\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}}$
$\Longrightarrow \ a^{2} = \ b^{2} + c^{2}(\because\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} = 0)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.