Vectors and Vector Products

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 $x$ , $y$ , and $z$ - 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:

$$\vA = A_x \hx + A_y \hy + A_z \hz$$

in terms of component scalar amplitudes $(A_x,A_y,A_z)$ and unit vectors in the orthogonal directions $(\hx,\hy,\hz)$ .

To add vectors (in Cartesian coordinates) we add components:

$$\vC = \vA + \vB = (A_x + B_x) \hx + (A_y + B_y) \hy + (A_z + B_z) \hz$$

The resultant is also the result of a geometric triangle or parallelogram rule:

vector-sum.eps

Subtraction is just addition of a negative:

$$\vC = \vA - \vB = (A_x - B_x) \hx + (A_y - B_y) \hy + (A_z - B_z) \hz$$

It can also be visualized by means of a geometric triangle so that $(\vA - \vB) + \vB = \vA$ .

vector-difference.eps