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

Answer 1

The correct option is B $\sqrt{\frac{2}{11}}$

$A(1,2,3),B(2,3,1),C(2,4,2),O(0,0,0)$
Equation of plane passing through $A,B,C$ is
$\left|\begin{array}{ccc}x-1& y-2& z-3\\ 2-1& 3-2& 1-3\\ 2-1& 4-2& 2-3\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}x-1& y-2& z-3\\ 1& 1& -2\\ 1& 2& -1\end{array}\right|=0$

$\Rightarrow (x-1)(-1+4)-(y-2)(-1+2)+(z-3)(2-1)=0$
$\Rightarrow 3(x-1)-(y-2)+(z-3)=0$
$\Rightarrow 3x-y+z-4=0$ is the required plane equation, whose normal $\overrightarrow{n}$ is $3\hat{i}-\hat{j}+\hat{k}.$
$\overrightarrow{OP}=2\hat{i}-\hat{j}+\hat{k}$
Projection of $\overrightarrow{OP}$ on plane
$=\text{Projection of}\overrightarrow{OP}\text{perpendicular to}\overrightarrow{n}$
$=|\overrightarrow{OP}\times \overrightarrow{n}|=\left|\right(2\hat{i}-\hat{j}+\hat{k})\times \frac{(3\hat{i}-\hat{j}+\hat{k})}{\sqrt{11}}|$
$=\left|\frac{+\hat{j}+\hat{k}}{\sqrt{11}}\right|=\sqrt{\frac{2}{11}}$