Vector Equations of Circles & Spheres

Vector equations of circles and spheres is part of the Vectors topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-02). The equation \(|\mathbf{r} - \mathbf{c}| = r\) says every point is a fixed distance \(r\) from the centre \(C\): in the plane it describes a circle, and in space a sphere.

Students and teachers will cover the parametric form \(\mathbf{r} = (r\cos\theta + x_C)\mathbf{i} + (r\sin\theta + y_C)\mathbf{j}\), converting to Cartesian equations, deciding whether a point lies inside, on or outside a sphere, and finding where a line meets a sphere in the HSC Extension 2 course.