Application of Vectors to 3-Dimensional Geometry

Vectors give an elegant, coordinate-free-until-you-need-it way to describe straight lines and planes in three dimensions. Every straight line and every plane is a subset of $\mathbb R^3$; from here on "line" always means "straight line."

A plane is a set $P$ of points in $\mathbb R^3$ such that whenever $A,B,C$ are any three non-collinear points of $P$, the line through any two of them lies entirely inside $P$. Two planes are intersecting if they share at least one point and also each has a point not on the other; coincident if they have exactly the same points; parallel but not coincident if they share no point at all. A line, similarly, can be understood as the set of points common to two intersecting planes. …