Q. The equation of the plane passing through three non-collinear points $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ is
A) $\overrightarrow{\mathbf{r}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) = 0$
B) $\overrightarrow{\mathbf{r}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) = \lbrack\overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}}\rbrack$
C) $\overrightarrow{\mathbf{r}} \bullet \left( \overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) \right) = \lbrack\overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}}\rbrack$
D) $\overrightarrow{\mathbf{r}} \bullet \left( \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) = 0$
Explanation
(b)
We know that the equation of the plane passing through three non-collinear points $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ is
$\overrightarrow{\mathbf{r}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) = \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.