Question

A unit vector perpendicular to the plane formed by the points $(1,0,1), (0, 2,2)$ and $(3,3, 0)$ is

• A. $\frac{1}{5\sqrt{3}}\left(5\hat{i}-\hat{j}-7\hat{k}\right)$
• B. $\frac{1}{5\sqrt{3}}\left(5\hat{i}-\hat{j}+7\hat{k}\right)$
• C. $\frac{1}{5\sqrt{3}}\left(5\hat{i}+\hat{j}+7\hat{k}\right)$
• D. None of these

Answer 1

Answer: (B)

Let $A(1, 0, 1), \,B(0,2,2)$ and $C (3,3, 0)$ be the given points, then
$\overrightarrow{AB} = -\hat{i}+2\hat{j}+\hat{k}, \overrightarrow{BC} = 3\hat{i}+\hat{j}-2\hat{k} $
$\overrightarrow{AB}\times \overrightarrow{BC} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ -1&2&1\\ 3&1&-2\end{vmatrix} = -5\hat{i}+\hat{j}-7\hat{k}$
$\therefore $ unit vector $\bot$ to the plane: ABC
$ = \pm\frac{1}{5\sqrt{3}}\left(-5\hat{i}+\hat{j}-7\hat{k}\right)$