Lines and planes in space

To describe a line in space you need a point it passes through and a direction to travel in — that is exactly a position vector plus a multiple of a direction vector, r = r₀ + tv. To describe a plane you need a point and, more usefully, a vector perpendicular to it: the normal. Everything else follows from those two descriptions.

The normal is the idea worth dwelling on, because it sticks out of the plane rather than lying in it. Its power is that a point lies in the plane exactly when the vector from the reference point to it is perpendicular to the normal — that is, when a dot product is zero. Expand that condition in coordinates and you get the familiar ax + by + cz = d, where (a, b, c) is the normal vector, readable straight off the equation.

Two warnings. A plane has infinitely many valid equations: scale the whole thing by 2 and it describes the same plane, since the normal's length was never the point, only its direction. And in three dimensions, two lines that are not parallel need not meet — they can pass each other at different heights, which is impossible on a flat page. These are skew lines, and forgetting they exist is the most common error carried over from two-dimensional intuition. Whether two lines are parallel, intersecting or skew is settled by a cross product and a dot product, not by drawing.