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 .