This definition does not seem too involved until you remember that forces are vector quantities. Adding vectors is not the same as adding scalars. You have to account for the direction of the forces involved. The following situations will show the general procedures followed when adding force vectors. If you think about the relative directions of the forces you are adding the process should become apparent. No individual step in any of the processes is difficult, but in some cases the procedure requires a combination of many steps.

For forces in the same direction

If two (or more) forces are in the same direction you can add the magnitudes of the forces together. The direction of the total force would be the same direction as the original forces.

EX. You push horizontally on a couch with a force of 200 N while your friend pulls on it
in the same direction with a force of100 N.

In this case the force diagram is a little overkill, but here it is with the solution.

For forces in opposite directions

If two (or more) forces are in opposite directions you first need to assign a pos/neg value to each force. Which is which does not matter as long as forces in the same direction have the same sign and forces in opposite directions have opposite signs.

Once the pos./neg. have been labeled you can add the magnitudes of the forces together. The direction of the total force would be determined by the sign of the result.

EX. You and an opponent each kick a soccer ball at the same time from opposite directions. Your force is 75 N and your opponent's kick is 125 N.

Again, the force diagram is not strictly required, but here it is with the solution.

For forces in perpendicular directions

If two forces are in opposite directions you first need to draw the force diagram to get the information correct. To add the vectors you must:

• construct a rectangle using the given vectors as two sides
• draw the diagonal of the rectangle starting at the same corner as the two given vectors (this represents the total vector)
• use the Pythagorean formula to calculate the magnitude of the total vector
• use the tangent function to calculate one of the angles formed by the total vector

Be sure to express the answer in similar terms as the original vectors.

This is a multistep process, don't shortcut it.

EX. An airplane heads due north in a region where the wind is blowing in an easterly direction.
If the plane's engines push it forward with a force of 2.5 E3 N while the wind's force is
1.0 E3 N, what is the total force on the plane?

Here are the steps of the solution:

For forces in nonperpendicular directions

This process is similar to adding forces which are perpendicular, but you must resolve each vector into its components first.

The process:

• draw the force diagram
• for any vector not on an axis, resolve the vector into its components
• add the vectors on an axis to get a subtotal for that axis
• construct a rectangle using the subtotal axis vectors as two sides
• draw the diagonal of the rectangle starting at the same corner as all the other vectors (this represents the total vector)
• use the Pythagorean formula to calculate the magnitude of the total vector
• use the tangent function to calculate one of the angles formed by the total vector

First, Draw the force diagram.

Second, resolve the boat's thrust into components.

Third, find the total X and Y components.

Fourth, use Pythagorean formula and Tangent function to find the magnitude and angle for the total force.

Last, express the answer in similar terms as the given information.

For addition of more than two forces

In cases where there are more than two forces the procedure to follow is the same as that for nonperpendicular forces. Just remember to resolve any force which is not on an axis into its components first.

In Summary: