Q.Find the Cartesian equations of the line passing through $A(2, 2, 1)$ and $B(1, 3, 0)$.

Given two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, the direction ratios of the line joining them are simply the coordinate differences $x_2-x_1,\ y_2-y_1,\ z_2-z_1$. Substituting these in place of $a,b,c$ in the point-with-direction-ratios formula gives the Cartesian equations of the line through two points: $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}.$ As with the direction-ratios form, this chain of equal fractions represents two independent equations, since a line in space needs two equations to be pinned down. This form is especially convenient for two closely related tasks: writing down the equation of a line once two of its points are known, and testing whether three given points are collinear — by writing the equation of the line through the fi …