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