Equations of Plane

A plane is a flat surface with the property that the line joining any two of its points lies entirely on it. A single plane can be pinned down by any one of several kinds of data: two intersecting lines, two parallel lines, a line together with a point that is not on it, or three points that are not collinear. All four of these situations turn up as different-looking theorems below, but they all reduce to the same underlying idea — once you know one point of the plane and one direction that is perpendicular to the plane, the plane is completely determined.

That perpendicular direction is called a normal to the plane. A given plane has infinitely many normal vectors, but every one of them is a scalar multiple of every other, so they all share the same direction ratios. Because of this, it is common (and harmless) to speak of "the" normal to a plane, even though strictly there are many parallel normal vectors — only the direction ratios matter, not the length or the sign.

As a first concrete example, the XY-plane (the plane containing the x and y axes) is normal to the z-axis, so the direction ratios of its normal are $0, 0, 1$. This simple fact is used later (Exercise 6.3, Q.7) to write the equation of any plane parallel to the XY-plane without any further computation. …