Q. Let $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ be the position vectors of the vertices $A,B,C$ respectively of $\mathrm{\Delta}\ ABC$. The vector area of $\mathrm{\Delta}\ ABC$ is
A) $\frac{1}{2}\left\{ \overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) + \overrightarrow{\mathbf{b}} \times \left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right) + \overrightarrow{\mathbf{c}} \times \left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) \right\}$
B) $\frac{1}{2}\left\{ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right\}$
C) $\frac{1}{2}\left\{ \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right\}$
D) $\frac{1}{2}\left( \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} \right)\ \overrightarrow{\mathbf{a}} + \left( \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right)\ \overrightarrow{\mathbf{b}} + (\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}})\overrightarrow{\mathbf{c}}$
Explanation
(b)
Area of triangle$= \frac{1}{2}\left\{ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right\}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.