Question

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

Answer 1

We know that the vector equation of the plane passing through a point $A\left(\overrightarrow{a}\right)$ and normal to $\overrightarrow{n}$ is $\overrightarrow{r}\cdot \overrightarrow{n}=\overrightarrow{a}\cdot \overrightarrow{n}$.

Here, $\overrightarrow{a}=3\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{n}=4\hat{i}+3\hat{j}+2\hat{k}$

$\therefore$ the equation of the required plane is

$\overrightarrow{r}\cdot (4\hat{i}+3\hat{j}+2\hat{k})=(3\hat{i}-2\hat{j}+\hat{k})\cdot (4\hat{i}+3\hat{j}+2\hat{k})$

$\Rightarrow \overrightarrow{r}\cdot (4\hat{i}+3\hat{j}+2\hat{k})=12-6+2$

$\Rightarrow \overrightarrow{r}\cdot (4\hat{i}+3\hat{j}+2\hat{k})=8$

This is the required vector equation.