How Do You Find the Point of Intersection of Two Lines?

Find the point of intersection by equating the corresponding $x$-, $y$-, and $z$-components of the two lines, solving for their parameters, and substituting back. A point exists only if the resulting parameter values satisfy all three component equations.

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 3.15: Classification of lines, and the method can be assessed on HL Papers 1, 2, or 3.

Each line is written in the vector equation of a line form $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where is a position vector to a known point, is a direction vector, and is a scalar parameter. At a common point, the two position vectors must be identical component by component.