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). This subtopic recognises \(|\mathbf{r} - \mathbf{c}| = r\) as the vector equation of a circle of radius \(r\) and centre \(C\) in two dimensions, and as the vector equation of a sphere in three dimensions.

Students and teachers will cover how to write a circle in the form \(\mathbf{r} = (r\cos\theta + x_C)\mathbf{i} + (r\sin\theta + y_C)\mathbf{j}\), and how to use these circle and sphere equations to solve problems — such as testing whether a point lies inside a sphere — in the HSC Extension 2 course.