Question

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

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

Answer 1

The correct option is D

$(1,-2,-1)$

The explanation for the correct option:

Step 1. Calculate the direction ratio:

Let the given points are$A(0,1,1)$, $B(1,1,2)$ and $C(-1,2,-2)$

The direction ratio of $AB$ $=\left[\right(1-0),(1-1),(2-1\left)\right]$ [$\because$ direction ratio of the points $\mathbf{(}\mathbf{}{\mathit{x}}_{\mathbf{1}\mathbf{}\mathbf{,}\mathbf{}}{\mathit{y}}_{\mathbf{1}\mathbf{}\mathbf{,}\mathbf{}}{\mathit{z}}_{\mathbf{1}}\mathbf{)}$ and $\mathbf{(}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{y}}_{\mathbf{2}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{z}}_{\mathbf{2}}\mathbf{)}$ is

$=(1,0,1)$ $\mathbf{}\mathbf{(}{\mathit{x}}_{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathit{x}}_{\mathbf{1}}\mathbf{)}\mathbf{}\mathbf{,}\mathbf{}\mathbf{(}{\mathit{y}}_{\mathbf{2}}\mathbf{-}{\mathit{y}}_{\mathbf{1}}\mathbf{)}\mathbf{}\mathbf{,}\mathbf{}\mathbf{(}{\mathit{z}}_{\mathbf{2}}\mathbf{-}\mathbf{}{\mathit{z}}_{\mathbf{1}}\mathbf{)}\mathbf{]}$

The direction ratio of $BC$ $=\left[\right(-1-1),(2-1),((-2-2)]$

$=(-2,1,-4)$

Step 2. Calculate the normal plane:

The normal to the plane will be perpendicular to both lines. let it be n, then

$n=ABxBC$

$$\begin{gathered} n=\left|\begin{array}{ccc}i& j& k\\ 1& 0& 1\\ -2& 1& -4\end{array}\right| \\ =i(0-1)-j(-4+2)+k(1-0) \\ =-i-j(-2)+k \\ =-i+2j+k \end{gathered}$$

Therefore, the direction ratio is $(-1,2,1)$ or $-(1,-2,-1)$

Hence, the correct option is (D).