Question

Direction ratios of the normal to the plane passing through the points $(0,1,1),(1,1,2)$ and $(-1,2,-2)$ are

• A. $(1,1,1)$
• B. $(2,1,-1)$
• C. $(1,2,-1)$
• D. $(1,-2,-1)$

Answer 1

The correct option is D $(1,-2,-1)$

$\overrightarrow{a}=0\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}+2\hat{k},\overrightarrow{c}=-\hat{i}+2\hat{j}-2\hat{k}$

$\therefore$ Direction of line $\overrightarrow{AB}$ and $\overrightarrow{BC}$

Directions of normal is perpendicular to that of lines in plane.

$\therefore \overrightarrow{n}=\overrightarrow{AB}\times \overrightarrow{BC}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 0& 1\\ -2& 1& -4\end{array}\right|=\widehat{-i}+2\hat{j}+\hat{k}$

$\therefore$ Directions of normal is $(1,-2,-1)$