 So now that we have a geometric interpretation of vectors as a directed distance, we can use these to describe equations for geometric objects like lines and planes, because these equations generally have some sort of parameter in them. We refer to them as parametric equations. Suppose I want to write the equation for a line PQ using vectors. So first of all, there's a natural vector that describes the direction of the line, which is the vector PQ. And so if I'm on the line, I could get to any other point by following that direction for some distance. And so this suggests the following approach. First, we need to get to a point P on the line. Once we're there, we can go in the direction of the line any distance to get to another point on the line. And what this suggests is the following for the vector equation for a line. All points x on a line PQ can be expressed as vector OP, getting from the origin to the point, plus T times vector PQ following the line some arbitrary distance. So if I want to write the parametric equation of the line through the points P31, negative 1, and Q1, 1, 1, 5, the first thing I want to do is I want to find the vector that points in the same direction as the line. I want to find the vector PQ. So we can find that by subtracting the coordinates of the beginning and ending points. And we get our vector PQ negative 2, 0, 6. The vector OP is going to have the same components as the coordinates of P. And now I have the vector that'll take me to a point on the line and the vector that'll take me along the line, so I can substitute those into our equation. So what about the parametric equation of a plane? Three points are enough to define a plane. And there's two natural vectors we can define going from one point to each of the other two points. Now it's worth also noting that any linear combination of these two points will take us to another point on the plane. And so this suggests a way that we can define the vector equation of a plane starting at the origin, get to a point on the plane, and then form any linear combination of the two vectors that are already in the plane. And this leads to the following vector equation. So we'll go from the origin to one of the known points on the plane, and then we'll add on any linear combination of two vectors in the plane. So if I want to write the vector equation of the plane through the points P, Q, and R, first I need to get to a point on the plane. For example, I can take the vector OP, and then I can add to that any linear combination of the two vectors that are in the plane, P, Q, and PR. And so this gives me the parametric or vector equation of the plane. Any point on the plane is this vector OP plus some linear combination of the two vectors in the plane.