Separating vectors into components

While sometimes it is useful to know just the magnitude and direction of a vector, other times (particularly when we’re doing calculations) it’s much more useful to know the horizontal and vertical components of a vector.

What are components?

Theyr’e just the horizontal and vertical parts of a vector.

So instead of describing a vector as “5m at 30 degrees to the horizontal”, we can describe it as “4.33m horizontally and 2.5m vertically”.

How to find the components of a vector

To find the components of a vector, we can just use some trigonometry.

For this vector:

        ^               +
        |             /               
        |           /                 
        |         /  F                
  Fcosθ |       /                     
        |     /                       
        |   /                         
        | /   θ                        
        + -------------->             
              Fcosθ    

For the vector indicated by the diagonal arrow with magnitude F:

• The horizontal component can be found using F \cos \theta
• The vertical component can be found using F \sin \theta
• \theta is the angle between the vector and the horizontal axis.

So if we know the magnitude of the vector and the angle it makes with the horizontal, we can easily find its components using these formulas.

Find the horizontal and vertical components of the vector 9m at 30^\circ to the horizontal.

• Horizontal component: 9 \cos 30^\circ = 9 \times 0.866 = 7.794m
• Vertical component: 9 \sin 30^\circ = 9 \times 0.5 = 4.5m

Uses of components of vectors

We need to split up vectors into their components whenever we do any calculations regarding motion.