Question

A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2)$. If $O$ is the origin and $P$ is (2,-1,1) , then the projection of $\overrightarrow{OP}$ on this plane is of length :

• A. $\sqrt{\frac{2}{7}}$
• B. $\sqrt{\frac{2}{3}}$
• C. $\sqrt{\frac{2}{11}}$
• D. $\sqrt{\frac{2}{5}}$

Answer 1

Answer: (C)

Normal to plane
$\vec{n}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -2\\ 0& 1& 1\end{array}\right|$
$=3\hat{i}-\hat{j}+\hat{k}$
$\overrightarrow{OP}=2\hat{i}-\hat{j}+\hat{k}$
$\cos \theta =\frac{6+1+1}{\sqrt{6}\sqrt{11}}=\frac{8}{\sqrt{66}}$
$\Rightarrow \sin \theta =\sqrt{\frac{2}{66}}$
$\therefore$ Projection of $\overrightarrow{OP}$ on plane $=|\overrightarrow{OP}|\sin \theta$
$=\sqrt{\frac{2}{11}}$