Parametrization of a plane

A plane is determined by a point p in the plane and two vectors a, and b parallel to the plane. (This is identical to the case with three points in the plane. Why?)

From this fact, we can parametrize a plane just like we parametrized a line (only we'll need two parameters instead of one). If x is a point in the plane, the vector from p to x (i.e., x - p) is some multiple of a plus some multiple of b. (Can you see why?) We can express this as x - p = sa + tb for t, s $\in$ R. Usually, we'll write this as x = p + sa + tb. The real numbers s and t are the parameters for this parametrization of the line plane.

The idea of the parametrization is that as the parameters s and t sweep through all real numbers, the point x sweeps out the plane. In other words, it is a two-dimensional analogue of the parametrization of the line.

Here we've added the point x in cyan. You can't move x directly, but you can move it by changing the parameters s and t (the blue points on sliders).

Clearly, for any value of s and t, the point x lies on the plane. Also, by changing s and t, you can move the point x to any position on the plane.