Q. The projection of point $P(\overrightarrow{p})$ on the plane $\overrightarrow{r} \bullet \overrightarrow{n} = q$ is $(\overrightarrow{s})$, then
A) $\overrightarrow{s} = \frac{(q - \overrightarrow{p} \bullet \overrightarrow{n})\overrightarrow{n}}{\left| \overrightarrow{n} \right|^{2}}$
B) $\overrightarrow{s} = \overrightarrow{p} + \frac{(q - \overrightarrow{p} \bullet \overrightarrow{n})\overrightarrow{n}}{\left| \overrightarrow{n} \right|^{2}}$
C) $\overrightarrow{s} = \overrightarrow{p} - \frac{(\overrightarrow{p} \bullet \overrightarrow{n} \bullet )\overrightarrow{n}}{\left| \overrightarrow{n} \right|^{2}}$
D) $\overrightarrow{s} = \overrightarrow{p} - \frac{(q - \overrightarrow{p} \bullet \overrightarrow{n})\overrightarrow{n}}{\left| \overrightarrow{n} \right|^{2}}$
Explanation
(b)
We have $\overrightarrow{s} - \overrightarrow{p} = \lambda\overrightarrow{n}$ and $\overrightarrow{s} \bullet \overrightarrow{n} = q$
$\Rightarrow \left( \lambda\overrightarrow{n} + \overrightarrow{p} \right) \bullet \overrightarrow{n} = q$
$\Rightarrow \ \lambda = \frac{q - \overrightarrow{p} \bullet \overrightarrow{n}}{\left| \overrightarrow{n} \right|^{2}}$
$\Rightarrow \ \overrightarrow{s} = \overrightarrow{p} + \frac{(q - \overrightarrow{p} \bullet \overrightarrow{n})\overrightarrow{n}}{\left| \overrightarrow{n} \right|^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.