Coordinates and vectors in space

Build spatial reasoning with coordinates and vectors in three dimensions. This precalculus extension practices coordinate planes, displacement, distance, midpoints, unit vectors and sphere equations. A labeled perspective diagram supports projection reasoning while explicitly warning that a flat sketch does not preserve spatial angles. Single-select and multiselect problems progress from component interpretation to geometric constraints and model checks. Detailed explanations connect every calculation with its meaning and distinguish a vector, a point, a sphere and a plane.