Straight Line Passing Through Two Given Points

Theorem 6.12. The line through two given points with position vectors $\vec a$ and $\vec b$ has parametric vector equation

This follows immediately from Theorem 6.11 by taking the direction to be $\vec b-\vec a$ (the vector from the first point to the second) instead of a separately-given $\vec b$.

(b) Non-parametric form: $(\vec r-\vec a)\times(\vec b-\vec a)=\vec 0$.

(c) Cartesian equations. With $P=(x,y,z)$, $A=(x_1,y_1,z_1)$, $B=(x_2,y_2,z_2)$, substituting into the parametric form and comparing components gives $x-x_1=t(x_2-x_1)$, etc., i.e.

$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}.$ …