Question

If position vectors of points $P,Q,R$ with respect to origin $O$ are $\overset{ \rightarrow }{r_{1}}=3\hat{i}-2\hat{j}-\hat{k},\overset{ \rightarrow }{r_{2}}=\hat{i}+3\hat{j}+4\hat{k}$ and $\overset{ \rightarrow }{r_{3}}=2\hat{i}+\hat{j}-2\hat{k}$ , then the distance between $P$ and plane $OQR$ is:-

• A. $1$
• B. $2$
• C. $3$
• D. None

Answer 1

Answer: (C)

Here, coordinates of $P\left(\right.3,-2,-1\left.\right),Q\left(\right.1,3,4\left.\right),R\left(\right.2,1,-2\left.\right)$ .
Equation of plane using three point form i.e. $\begin{vmatrix} x-0 & y-0 & z-0 \\ 1-0 & 3-0 & 4-0 \\ 2-0 & 1-0 & -2-0 \end{vmatrix}=0$
i.e. $2x-2y+z=0$ .
Now, distance of $P$ from this plane $=\frac{\left|2 \left(\right. 3 \left.\right) - 2 \left(\right. - 2 \left.\right) + \left(\right. - 1 \left.\right)\right|}{\sqrt{2^{2} + \left(\right. - 2 \left.\right)^{2} + 1^{2}}}=3$ units