Point of Intersection of Two Straight Lines

Given two lines $\dfrac{x-x_1}{a_1}=\dfrac{y-y_1}{a_2}=\dfrac{z-z_1}{a_3}$ and $\dfrac{x-x_2}{b_1}=\dfrac{y-y_2}{b_2}=\dfrac{z-y_2}{b_3}$, every point on the first is of the form $(x_1+sa_1,\ y_1+sa_2,\ z_1+sa_3)$ and every point on the second is $(x_2+tb_1,\ y_2+tb_2,\ z_2+tb_3)$, for parameters $s,t$.

Method. If the lines actually intersect, some common point must exist — so equate the two general points coordinatewise, giving three scalar equations in the two unknowns $s,t$:

Solve any two of these three equations for $s$ and $t$. Then check whether these values also satisfy the third, remaining equation:

• if yes, the lines genuinely intersect, and substituting the found value of $s$ (or $t$) back into the corresponding general point gives the point of intersection; …