Q. If $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}$ are unit vectors, then $\left| \ \overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{b}}\ \left. \ \right|^{2} + \right|\overrightarrow{\mathbf{b}}\mathbf{-}\overrightarrow{\mathbf{c}}\ \left| \ ^{2} + |\overrightarrow{\mathbf{c}}\mathbf{-}\overrightarrow{\mathbf{a}} \right|^{2}$ does not exceed
A) 4
B) 9
C) 8
D) 6
Explanation
(b)
Given expression $= 2(1 + 1 + 1) - 2\sum_{}^{}(\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}})$
$= 6 - 2\sum_{}^{}(\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}})$ …(i)
But $\left( \overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}} \right)^{2} \geq 0$
$\therefore(1 + 1 + 1) + 2\sum_{}^{}{\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}} \geq 0$
$\therefore 3 \geq - 2\sum_{}^{}{\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}}$ …(ii)
From relations (i) and (ii), we get
Given expression $\leq 6 + 3 = 9$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.