Question

Find unit vector perpendicular to the plane determined by $P\left(1,-1,2\right),Q\left(2,0,-1\right)$ and $R\left(0,2,1\right)$

Answer 1

$P=(1,-1,2),Q=(2,0,-1),R=(0,2,1)$

$PQ=\overrightarrow{a}=\hat{i}+\hat{j}-3\hat{k}=\overrightarrow{OB}-\overrightarrow{OA}$

$QR=\overrightarrow{b}=-2\hat{i}+2\hat{j}+2\hat{k}$

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -3\\ -2& 2& 2\end{array}\right|$

$=8\hat{i}+4\hat{j}+4\hat{k}$

$\overrightarrow{a}=\left|\overrightarrow{a}\right|\hat{a}$

$\hat{a}=\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}=\frac{8\hat{i}+4\hat{j}+4\hat{k}}{4\sqrt{6}}$

$=\pm \frac{2\hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}$