Resolution of a Vector — Rectangular and Non-Rectangular Components

Just as two vectors can be combined into a single resultant vector, a single vector can be

split, or resolved, into two (or more) component vectors along chosen directions, such

that the vector sum of the components reproduces the original vector exactly. This is the

reverse operation to vector addition, and it is the single most useful vector technique in

all of mechanics, because it turns any vector problem into ordinary algebra along convenient

axes.

Rectangular resolution. The most common choice is to resolve a vector $\vec A$ along a

pair of mutually perpendicular axes, conventionally the x-axis and y-axis. If $\vec A$ makes

an angle $\theta$ with the positive x-axis, its rectangular components are

and the original vector can always be recovered from these components:

The vector $\vec A$, together with its two rectangular components, forms a right-angled

triangle (see the figure), so the relation $A^2 = A_x^2 + A_y^2$ is simply the Pythagorean

theorem applied to that triangle.

Non-rectangular resolution. A vector can equally well be resolved along two directions

that are not perpendicular to each other — for instance, along two given, non-orthogonal

reference lines in the plane. In that case the parallelogram law is applied in reverse:

$\vec A$ is taken as the diagonal of a parallelogram whose two adjacent sides lie along the

two chosen (non-perpendicular) directions, and the sine rule is used, with the known angle …