Components of a Vector

Choosing a coordinate system lets any vector be split into perpendicular pieces along the axes — its components — turning vector problems into ordinary (scalar) algebra performed three times over, once per axis.

In three dimensions, a vector $\vec A$ is written in component form as

where $A_x$, $A_y$, $A_z$ are the (signed) components of $\vec A$ along the $x$-, $y$- and $z$-axes.

In two dimensions, if $\vec A$ makes angle $\theta$ with the $x$-axis, its components are read straight off a right triangle with $\vec A$ as hypotenuse:

$A_x = A\cos\theta, \qquad A_y = A\sin\theta$ …