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 
.
