Question

The direction ratios of two lines AB, AC are 1, -1, -1 and 2, -1, 1. The direction ratios of the normal to the plane ABC are

• A. $2,3,-1$
• B. $2,2,1$
• C. $3,2,-1$
• D. $-1,2,3$

Answer 1

The correct option is A $2,3,-1$

Given : Direction ratios of $AB$ and $AC$ are $1,-1,-1$ and $2,-1,1$

i.e. $\overrightarrow{AB}=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{AC}=2\hat{i}-\hat{j}+\hat{k}$
$\therefore$ Normal vector to the plane ABC is given by
$=\overrightarrow{AC}\times \overrightarrow{AB}$
$=(2\hat{i}-\hat{j}+\hat{k})\times (\hat{i}-\hat{j}-\hat{k})$

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

$=2\hat{i}+3\hat{j}-\hat{k}$
Hence, the direction ratios of normal are $2,3,-1$