Vectors can be multiplied in two ways:

1. The scalar product gives product of a vector and the projection of  the other vector upon the first one. This is calculated according to

a b = ab cos(v)

The result is a scalar.  This statement can be proved with the following calculation:

Let C= A+B and form

C C = (A+B) (A+B) =

A² + B²+2AB

Solving for AB =(C²-A²-B²/2

which is a scalar quantity since it is made up of absolute values.

An example is the amount of work, W, done by operating against a force F a distance x.

W=F ×cos(v)

where v is the angle between F and  displacement x.

2. As the vectorproduct

a x b = ab sin(v).

This gives the area of the parallellogram formed by vectors a and b. It can be shown (it follows directly by computing the vectorproduct of  (a1,a2,a3) and  (b1,b2,b3) and Sarrus rule) that the vectorproduct is another vector forming an orthogonal coordinatesystem with a and b.

Ex The Lorentzian force in physics is given by

F =q vxB. 

Here F equals the force on the particle with charge q moving with speed v through the magnetic field B.