Find the equation of the line in vector and in Cartesian form that passes through the point with position vector
Question:

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector $2 \hat{i}-\hat{j}+4 \hat{k}$ and is in the direction $\hat{i}+2 \hat{j}-\hat{k}$.

Solution:

It is given that the line passes through the point with position vector

$\vec{a}=2 \hat{i}-\hat{j}+4 \hat{k}$                    …(1)

$\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$                    $\ldots(2)$

It is known that a line through a point with position vector $\vec{a}$ and parallel to $\vec{b}$ is given by the equation, $\vec{r}=\vec{a}+\lambda \vec{b}$

$\Rightarrow \vec{r}=2 \hat{i}-\hat{j}+4 \hat{k}+\lambda(\hat{i}+2 \hat{j}-\hat{k})$

This is the required equation of the line in vector form.

$\vec{r}=x \hat{i}-y \hat{j}+z \hat{k}$

$\Rightarrow x \hat{i}-y \hat{j}+z \hat{k}=(\lambda+2) \hat{i}+(2 \lambda-1) \hat{j}+(-\lambda+4) \hat{k}$

Eliminating λ, we obtain the Cartesian form equation as

$\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-4}{-1}$

This is the required equation of the given line in Cartesian form.