### Resolving Vectors into Axial Components

We know that when two vectors are in the same dimension, they can be added arithmetically. Suppose we have two vectors that are on a north-south, east-west grid, as shown below. One of the methods we can use to add these vectors is to resolve each one into a pair of vectors that lay on the north-south and east-west axes.

The two vectors we are to add is a force of 65 N at 30° north of east and a force of 35 N at 60° north of west.

We can resolve each of the vectors into two components on the axes lines. Each vector is resolved into a component on the north-south axis and a component on the east-west axis.

Using trigonometry, we can resolve (break down) each of these vectors into a pair of vectors that lay on the axial lines (shown in red above).

The east-west component of the first vector is (65 N)(cos 30° ) = (65 N)(0.866) = 56.3 N east

The north-south component of the first vector is (65 N)(sin 30°) = (65 N)(0.500) = 32.5 N north

The east-west component of the 2^{nd} vector is (35 N)(cos 60°) = (35 N)(0.500) = 17.5 N west

The north-south component of the 2^{nd} vector is (35 N)(sin 60°) = (35 N)(0.866) = 30.3 N north

### Summary

- Vectors can be resolved into component vectors that lie on the axes lines.

### Review

- A force of 150. N is exerted 22° north of east. Find the northward and eastward components of this force.
- An automobile travels a displacement of 75 km 45° north of east. How far east does it travel and how far north does it travel?

### Explore More

Use this resource to answer the questions that follow.

- What does SohCahToa mean?
- Why is SohCahToa relevant to resolving a vector into components?
- Why is the sum of the components larger than the resultant vector?