Question

Vector equation of the plane passing through a point having position vector $2\hat{i}+3\hat{j}-4\hat{k}$ and perpendicular to the vector $2\hat{i}-\hat{j}+2\hat{k}$ is

• A. $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})+7=0$
• B. $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=7$
• C. $\overrightarrow{r}\cdot (-3\hat{i}-2\hat{j}-3\hat{k})=0$
• D. $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=9$

Answer 1

The correct option is A $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})+7=0$

Using $(\overrightarrow{r}-\overrightarrow{a})\cdot \overrightarrow{n}=0$
i.e.$\overrightarrow{r}\cdot \overrightarrow{n}=\overrightarrow{a}\cdot \overrightarrow{n}$
$\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=(2\hat{i}+3\hat{j}-4\hat{k})\cdot (2\hat{i}-\hat{j}+2\hat{k})$
$\Rightarrow \overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=4-3-8$
$\Rightarrow \overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=-7$
Hence option (a) is correct.