# Components of a Vector

In a two-dimensional coordinate system, any vector can be broken into x -component and y -component.

For example, in the figure shown below, the vector is broken into two components, vx and vy . Let the angle between the vector and its x -component be θ.

The vector and its components form a right angled triangle as shown below.

In the above figure, the components can be quickly read. The vector in the component form is .

The trigonometric ratios give the relation between magnitude of the vector and the components of the vector.

vx = v cos θ

vy = v sin θ

Using the Pythagorean Theorem in the right triangle with lengths vx and vy:

Here, the numbers shown are the magnitudes of the vectors.

Case 1: Given components of a vector, find the magnitude and direction of the vector.

Use the following formulas in this case.

Magnitude of the vector is .

To find direction of the vector, solve for θ.

Case 2: Given the magnitude and direction of a vector, find the components of the vector.

Use the following formulas in this case.

vx = v cos θ

vy = v sin θ

Example:

The magnitude of a vector is 10 units and the direction of the vector is 60° with the horizontal. Find the components of the vector.

F x = F cos 60°

= 5

F y = F sin 60°

So, the vector is .