Vector diagrams

Two forces can be added together to find a resultant force.

In the diagram of a toy trailer below, when two children pull on the handle, one with a force of 3 newtons (N) to the left and one with a force of 4 newtons (N) to the right, the combined effect of the two forces is to pull the trailer to the right with a resultant force of 4 – 3 = 1 newton (N).

A red cart has a red arrow representing 3 Newtons pointing left and a green arrow representing 4 Newtons pointing right.

Example

What would be the resultant force on the trailer if both children pulled to the right?

Resultant force = 4 + 3 = 7 N to the right

In the diagram of a toy trailer below, when two children pull on the handle, one vertically with a force of 3 newtons (N) and one horizontally with a force of 4 newtons (N), the combined effect of the two forces is to pull the trailer upwards away from the ground and to the right.

A red cart has a red arrow representing 3 Newtons pointing up and a green arrow representing 4 Newtons pointing right.

The sum of these two forces can be found using a vector diagram. The two forces are represented by the adjacent sides of a parallelogram, the lengths of the sizes being proportional to the size of the forces.

The resultant force is represented in size and direction by the diagonal of the parallelogram.

Draw a parallelogram to scale, in which two adjacent sides each represent a force. Try to choose a simple scale, for example 1 cm = 1 N. For the toy trailer example above, draw:

  • a horizontal line representing the 4 N force
  • a vertical line representing the 3 N force
  • arrow heads to show the direction in which each force acts
  • complete the parallelogram
  • measure the length of the diagonal
  • use the same scale to convert this length to the resultant force
  • measure the angle of the resultant force to the horizontal
A toy trailer has a handle in a position as if it is being pulled, with an arrow on the end of the handle.

The diagonal of the parallelogram shows that the resultant force is 5 newtons (N) acting at an angle of 37° to the horizontal.