Resolving Vectors into Components

Resolving Vectors into Components

We have just shown that any two vectors can be represented by a single resultant vector that has the same effect. Guess what?! You can do the same thing in reverse! Any single vector can be represented by two other vectors (components), which would have the same effect as the original one:

Resolving Vectors into Components

You need to use trigonometry to find the two components of a vector. Remember the two components will always be at right angles.

Resolving Vectors into Components

Check that you understand how to calculate the values of the components.

(If trigonometry is your worst nightmare, you can always draw scale drawings instead and then measure the components off the diagram.)

Worked Example 1:

Find the forward force on the boat.

Resolving Vectors into Components

First you need to find the amount of the 5 N force that acts in the forward direction, using trigonometry:

Part of 5 N force in forward direction = 5 cos 30° = 4.3 N

Then this can be added to the 7 N force:

4.3 + 7 = 11.3 N force in the forward direction.

Worked Example 2:

Sometimes the direction we are interested in is not vertical or horizontal. It doesn't matter as long as we still only add parallel forces.

Find the force on the car parallel to the slope:

Resolving Vectors into Components

The 2000 N force is already parallel to the slope so we can ignore it for a moment.

The 10 000 N is at an angle of 60 degrees to the slope so we need to use trigonometry to find its component parallel to the slope (look at the small triangle carefully):

Component parallel to the slope = 10 000 cos 60° = 5 000 N down the slope.

Now we can simply subtract the 2 000 N from the 5 000 N force as they are in opposite directions.

So the resultant force parallel to the slope = 5 000 - 2 000 = 3 000 N down the slope.

Now it's your turn!


Resolving Vectors into Components
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Hint: Resolve the force of the wind into its horizontal and vertical components, before trying to add the forces together. Remember you can't add forces that are perpendicular, you'll have to use Pythagoras!