A vector is a quantity with dimensions, a magnitude, and
a direction relative to a specific coordinate frame. Note
that it isn't sufficient to have a list of (say) three numbers labelled
,
, and
- the components have to transform when the
underlying coordinate frame is transformed ``like a vector''. Although
there are multiple coordinate systems in which vectors can be expressed,
the ``simplest'' one is Cartesian, where a vector can typically be
written:
in terms of component scalar amplitudes
and unit vectors
in the orthogonal directions
.
To add vectors (in Cartesian coordinates) we add components:
The resultant is also the result of a geometric triangle or parallelogram rule:
Subtraction is just addition of a negative:
It can also be visualized by means of a geometric triangle so that
.