Question

The image of the point with position vector $\hat{i}+3\hat{k}$ in the plane $\overrightarrow{r}.(\hat{i}+\hat{j}+\hat{k})=1$ is

• A. $\hat{i}+2\hat{j}+\hat{k}$
• B. $\hat{i}+2\hat{j}-\hat{k}$
• C. $-\hat{i}-2\hat{j}+\hat{k}$
• D. $\hat{i}-2\hat{j}+\hat{k}$

Answer 1

The correct option is C

$-\hat{i}-2\hat{j}+\hat{k}$

Explanation for correct option:

Step-1 Coordinate of point lie on given plane:

Point $P$ is given by $\hat{i}+3\hat{k}$.

So the co-ordinate of $P$ is $\left(1,0,3\right)$

The equation of the line $PR$ with direction ratio$\left(1,1,1\right)$ is

$$\begin{gathered} \overrightarrow{r}=\hat{i}+3\hat{k}+\lambda \left(\hat{i}+\hat{j}+\hat{k}\right) \\ =\left(1+\lambda \right)\hat{i}+\lambda \hat{j}+\left(3+\lambda \right)\hat{k} \end{gathered}$$

If this are the co-ordinates of point $R$, then

$$\begin{gathered} \because \overrightarrow{r}.(\hat{i}+\hat{j}+\hat{k})=1 \\ \Rightarrow \left(\left(1+\lambda \right)\hat{i}+\lambda \hat{j}+\left(3+\lambda \right)\hat{k}\right).(\hat{i}+\hat{j}+\hat{k})=1 \\ \Rightarrow 1+\lambda +\lambda +3+\lambda =1 \\ \Rightarrow 3\lambda =-3 \\ \Rightarrow \lambda =-1 \end{gathered}$$

Therefore the co-ordinate of $R$ are $\left(1+\lambda ,\lambda ,3+\lambda \right)=\left(0,-1,2\right)$

Step-2 Image of a point on a plane:

$R$ is the mid point of $PQ$.

Let co-ordinate of $Q$ be $\left(x,y,z\right)$, then

$\frac{x+1}{2}=0,\frac{y+0}{2}=-1,\frac{z+3}{2}=2$

or $\left(x,y,z\right)=\left(-1.-2,1\right)$

Image of the point $\hat{i}+3\hat{k}$ is $-\hat{i}-2\hat{j}+\hat{k}$

Hence, option C is correct.