Cambridge O Level Mathematics (Syllabus D) · Syllabus 4024 · Transformations and Vectors
Vector Geometry
What is Vector Geometry?
Vector geometry is the use of directed line segments and position vectors to settle geometric questions by calculation instead of by measurement. Fixing an origin O gives every point A a position vector OA, usually written a, and any vector between two points is then the difference of their position vectors, so AB equals b minus a. An unknown vector is found by choosing a route from its start point to its end point built only from vectors that are already known, since every valid route gives the same answer. The midpoint M of AB has position vector one half of a plus b, and a point P dividing AB internally in the ratio m to n has position vector n a plus m b all over m plus n, a result derived by travelling to A and then m over m plus n of the way along AB rather than quoted. Two non-zero vectors are parallel exactly when one is a scalar multiple of the other, and three points A, B and C are collinear exactly when AB is a scalar multiple of AC, because the two vectors are then parallel and share the point A. A parallelogram is established by showing one pair of opposite sides is equal as vectors, which fixes both length and direction, since equal magnitudes alone would not. The size and sign of the scalar also give relative positions and length scale factors, which is how vectors settle questions about ratio and similarity.
This definition is part of the Transformations and Vectors chapter in Cambridge O Level Mathematics (Syllabus D).