What is a vector?

An object used to describe a displacement

In geometry, a vector denoted ->AB (an arrow from point A to point B) represents a displacement from A to B. It is characterised by three elements:

• a direction: the line (AB)
• a direction: from A to B
• a magnitude, denoted ||AB||, which is the distance AB

Equal vectors

Two vectors are equal if they have the same direction, the same sense and the same magnitude, even if they do not lie on the same line. We say that they have different representations but represent the same vector.

Example: if ABDC is a parallelogram (note the order of the vertices), then ->AB = ->CD.

The zero vector

The zero vector, denoted ->0, is the vector whose origin and endpoint coincide (A = B). Its magnitude is 0; it has neither a defined direction nor a defined sense.

Common pitfall

Do not confuse ->AB and ->BA: they have the same magnitude and the same direction, but opposite senses. We write ->BA = -->AB.

Element	->AB	->BA
Direction	(AB)	(AB)
Sense	from A to B	from B to A
Magnitude	AB	AB