1. Tardigrade
  2. Question
  3. Mathematics
  4. 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 OP on this plane is of length :

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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 :

JEE MainJEE Main 2021Three Dimensional Geometry

Solution:

image
Normal to plane
$\vec{n} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 1&1&-2\\ 0&1&1\end{vmatrix}$
$=3 \hat{ i }-\hat{ j }+\hat{ k }$
$\overrightarrow{ OP }=2 \hat{ i }-\hat{ j }+\hat{ k }$
$\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 $=|\overrightarrow{ OP }| \sin \theta$
$=\sqrt{\frac{2}{11}}$