Components of a Vector
In a two-dimensional coordinate system, any vector can be broken into -component and -component.
For example, in the figure shown below, the vector is broken into two components, and . Let the angle between the vector and its -component be .
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.
Using the Pythagorean Theorem in the right triangle with lengths and :
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 .
Case 2: Given the magnitude and direction of a vector, find the components of the vector.
Use the following formulas in this case.
Example:
The magnitude of a vector is units and the direction of the vector is with the horizontal. Find the components of the vector.
So, the vector is .