Vector operations

Sum of two vectors

To add two vectors ->u and ->v, we can use the parallelogram rule or, more simply, Chasles’s relation:

->AB + ->BC = ->AC

This relation allows us to simplify vector sums, regardless of the intermediate point chosen.

Example: ->AB + ->BC + ->CD = ->AD

Multiplication by a real number

If k is a real number and ->u is a vector, then k->u is a vector:

• with the same direction as ->u
• with the same sense as ->u if k > 0, and with the opposite sense if k < 0
• with magnitude |k| * ||u||

If k = 0, then k->u = ->0.

Collinear vectors

Two non-zero vectors ->u and ->v are collinear if there exists a real number k such that ->v = k->u. This means that they have the same direction (lie on parallel or coincident straight lines).

This concept allows us to prove that three points A, B, C lie on a straight line: it suffices to show that ->AB and ->AC are collinear.

Common pitfall

Do not confuse ‘collinear’ (same direction) with ‘equal’ (same direction, same sense, same magnitude). Two collinear vectors may have opposite senses or different magnitudes.