Components and resultants of forces

A force has a size (its magnitude, in newtons) and a direction. Quantities like this are called vectors.

Two 10 N forces pushing the same way add to 20 N. Two 10 N forces pushing opposite ways cancel to 0. So you can't just add the sizes; you have to take directions into account.

The trick is to split every force into a horizontal part and a vertical part. Those can be added like ordinary numbers.

Suppose a force $P$ acts at angle $\theta$ above the horizontal. Draw it as the long side (hypotenuse) of a right-angled triangle, with the horizontal and vertical components as the other two sides.

From trigonometry (SOH CAH TOA):

• the side next to the angle is $P\cos\theta$;
• the side opposite the angle is $P\sin\theta$.

So:

$\text{horizontal component} = P\cos\theta, \qquad \text{vertical component} = P\sin\theta$

Careful: this is for $\theta$ measured from the horizontal. If the angle is measured from the vertical, the cos and sin swap.