components of vectors

Vector Components:

Vectors are generally oriented on a coordinate system, the most popular of which is the two-dimensional Cartesian plane. The Cartesian plane has a horizontal axis which is labeled x and a vertical axis labeled y. Some advanced applications of vectors in physics require using a three-dimensional space, in which the axes are x, y, and z.

This article will deal mostly with the two-dimensional system, though the concepts can be expanded with some care to three dimensions without too much trouble.

Vectors in multiple-dimension coordinate systems can be broken up into their component vectors. In the two-dimensional case, this results in a x-component and a y-component. The picture to the right is an example of a Force vector (F) broken into its components (F_x & F_y). When breaking a vector into its components, the vector is a sum of the components:

F = F_x + F_y

To determine the magnitude of the components, you apply rules about triangles that are learned in your math classes. Considering the angle theta (the name of the Greek symbol for the angle in the drawing) between the x-axis (or x-component) and the vector. If we look at the right triangle that includes that angle, we see that F_x is the adjacent side, F_y is the opposite side, and F is the hypotenuse. From the rules for right triangles, we know then that:

F_x / F = cos theta and F_y / F = sin theta   

which gives us

 Fx = F cos theta and Fy = F sin theta
Note that the numbers here are the magnitudes of the vectors. We know the direction of the components, but we're trying to find their magnitude, so we strip away the directional information and perform these scalar calculations to figure out the magnitude.

Further application of trigonometry can be used to find other relationships (such as the tangent) relating between some of these quantities, but I think that's enough for now.

Magnitude of the vector is .

To find direction of the vector, solve  for θ.