Q.Find the Cartesian equation of the line passing through the points $(1,-2,3)$ and $(4,0,-3)$.

A line through a point with position vector $\vec a$, parallel to a vector $\vec b$, has vector equation $\vec r=\vec a+\lambda\vec b$; equating components gives the Cartesian symmetric form $\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$, where $a,b,c$ are the line's direction ratios. When two points are given instead of a point and a direction, the direction ratios are simply the coordinate differences between them, and either point may serve as the base point $(x_1,y_1,z_1)$.