3.7.3—Position vectors

The position vector of P is OP, the directed vector from the origin O to P. The displacement from A to B is OB−OA, and the midpoint is the average of the two position vectors.

Choose one origin, preserve direction in subtraction, and translate geometric statements into vector equations such as a point dividing AB in a given ratio.

If OA=(2,1) and OB=(8,4), then AB=(6,3) and the midpoint has position (5,2.5).

AB is not the same as BA; reversing the order changes the sign of the displacement.