An ordinary number that does not change when the coordinate frame
changes is called a scalar. Multiplication of a vector by a
scalar re*scales* the vector by multiplying each of its components
as a special case of this rule:

Note well that the vector components
are themselves
scalars. Indeed, we build a vector in the first place by taking a
unit vector (of length one, ``pure direction'') and
*scaling* it by its component length, e.g.
, and then
summing the vectors that make up its components!

The multiplication of a vector by a scalar is commutative:

and distributive.

Robert G. Brown 2017-07-11