Q. If $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are three non-zero, non-coplanar vectors and ${\overrightarrow{\mathbf{b}}}_{1} + \overrightarrow{\mathbf{b}} - \frac{\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{a}}|}^{2}}\overrightarrow{\mathbf{a}}\ \ \ \ \ \ \ ,{\overrightarrow{\mathbf{b}}}_{2} + \overrightarrow{\mathbf{b}} - \frac{\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{a}}|}^{2}}\overrightarrow{\mathbf{a}}\ \ \$ And $\ {\overrightarrow{\mathbf{c}}}_{1} + \overrightarrow{\mathbf{b}} - \frac{\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{a}}|}^{2}}\overrightarrow{\mathbf{a}} + \frac{\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{b}}}{|{\overrightarrow{\mathbf{b}}|}^{2}}{\overrightarrow{\mathbf{b}}}_{1}\$ ${\overrightarrow{\mathbf{c}}}_{2} + \overrightarrow{\mathbf{c}} - \frac{\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{a}}|}^{2}}\overrightarrow{\mathbf{a}} - \frac{\overrightarrow{\mathbf{c}} \bullet {\overrightarrow{\mathbf{b}}}_{1}}{|{{\overrightarrow{\mathbf{b}}}_{1}|}^{2}}{\overrightarrow{\mathbf{b}}}_{1}\ ,\ \ \ \ \$ ${\overrightarrow{\mathbf{c}}}_{3} = \overrightarrow{\mathbf{c}} - \frac{\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{c}}|}^{2}}\overrightarrow{\mathbf{a}} - \frac{\overrightarrow{\mathbf{c}} \bullet {\overrightarrow{\mathbf{b}}}_{2}}{|\overrightarrow{\mathbf{c}}|^{2}}{\overrightarrow{\mathbf{b}}}_{1},\ \$ $\ {\overrightarrow{\mathbf{c}}}_{4} = \overrightarrow{\mathbf{c}} - \frac{\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{c}}|}^{2}}\overrightarrow{\mathbf{a}} - \frac{\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}}}{|\overrightarrow{\mathbf{b}}|^{2}}{\overrightarrow{\mathbf{b}}}_{1}$ Then, which of the following is a set of mutually orthogonal vectors?

A) $\left\{ \overrightarrow{\mathbf{a}},\ {\overrightarrow{\mathbf{b}}}_{1},{\overrightarrow{\mathbf{c}}}_{1} \right\}\$
B) $\left\{ \overrightarrow{\mathbf{a}},\ {\overrightarrow{\mathbf{b}}}_{1},{\overrightarrow{\mathbf{c}}}_{2} \right\}$
C) $\left\{ \overrightarrow{\mathbf{a}},\ {\overrightarrow{\mathbf{b}}}_{2},{\overrightarrow{\mathbf{c}}}_{3} \right\}$
D) $\left\{ \overrightarrow{\mathbf{a}},\ {\overrightarrow{\mathbf{b}}}_{2},{\overrightarrow{\mathbf{c}}}_{4} \right\}$
Explanation

(b)
$\overrightarrow{\mathbf{a}} \bullet {\overrightarrow{\mathbf{b}}}_{1} + \overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} - \frac{\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}}}{|{\overrightarrow{\mathbf{a}}|}^{2}}\overrightarrow{\mathbf{a}}\ \right)\ \$
$= \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} - \frac{\left| \overrightarrow{\mathbf{a}} \right|^{2}\left( \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}} \right)}{|\overrightarrow{\mathbf{a}}|^{2}}$
=$\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}} = 0$
Similarly, $\overrightarrow{\mathbf{a}} \bullet {\overrightarrow{\mathbf{c}}}_{2} = {\overrightarrow{\mathbf{b}}}_{1} \bullet {\overrightarrow{\mathbf{c}}}_{2} = 0$
Hence, $\{\overrightarrow{\mathbf{a}},{\overrightarrow{\mathbf{b}}}_{1},{\overrightarrow{\mathbf{c}}}_{2}\}$ are mutually orthogonal vectors.

Dr. Abubacker
Reviewed by Dr. Abubacker

Physics Educator & NEET/JEE Specialist

Ph.D. in Physics with 19+ years of teaching experience.

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