Vectors and Plane Geometry

In this section we apply the algebra of two-dimensional vectors to plane geometry. Given a point P(p₁, p₂) in the plane, the position vector of P is the vector P from the origin to P (Figure 10.2.1). P has components p₁ and p₂, so

P = p₁i + p₂j.

Figure 10.2.1: The position vector

If A and B are two points in the plane with position vectors A and B, then the vector from A to B is the vector difference B - A. This can be seen from Figure 10.2.2.

Figure 10.2.2

In Section 1.3, we saw that a line in the plane may be defined as the graph of.a linear equation

ax + by = c

where a and b are not both zero (Figure 10.2.3). We shall call the above equation a scalar equation of the line.

Figure 10.2.3

Figure 10.2.4

The position vector of any point P on a line L is called a position vector of L. If P and Q are two distinct points on L, the vector D from P to Q is called a direction vector of L. Thus D = Q - P (Figure 10.2.4).

Theorem 1 will show how to represent a line by a vector equation. Let us use the symbol X for the variable point X(x, y), and the symbol X for the variable vector X = xi + ,yj. (see Figure 10.2.5).

Figure 10.2.5