Question

Find the vector equation of the plane passing through $(1,2,3)$ and perpendicular to the plane $\overrightarrow{r}.(\hat{i}+2\hat{j}-5\hat{k})+9=0$

Answer 1

The position vector of the point $(1,2,3)$ is $\overrightarrow{r_1}=\hat{i}+2\hat{j}+3\hat{k}$
The direction ratios of the normal to the plane $\overrightarrow{r}.(\hat{i}+2\hat{j}-5\hat{k})+9=0$, are $1,2$ and $-5$ and the normal vector is $\overrightarrow{N}=\hat{i}+2\hat{j}-5\hat{k}$
The equation of a line passing through a point and perpendicular to the given plane is given by, $\overrightarrow{r}=\overrightarrow{r_1}+\lambda \overrightarrow{N},$
$\lambda \in R$
$\Rightarrow$ $\overrightarrow{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}+2\hat{j}-5\hat{k})$