Equation of a Circle Using Plane Vectors

Let be the center of a circle on the coordinate plane, and let the radius of the circle be . Let be any point on the circle. The distance between points and is given by: If the position vectors of points and are denoted as and respectively, then . Therefore, the equation becomes:

Conversely, any vector that satisfies represents a point that lies on the circle with center and radius .

Squaring both sides, we get:

Thus, the equation becomes:

Now, let be the center of the circle and be any point on the circle. The position vectors are and , so:

This is the equation of a circle with center and radius .