In many questions you will be told to add two or more vectors.

• Your first reaction might be to just take the two numbers and add them (like you would add 3 + 2 = 5).
• You absolutely positively can not do this!!! What you must remember is that vectors will most of the time be pointing in different directions at weird angles. You must "add" them using trigonometry and have a direction at the end.

Adding A + B

Let's look at a really simple example of adding vectors. For now we won't even have any numbers, just drawn out vectors named A and B.

Figure 1

If you were asked to add A and B, you would need to first arrange them as a diagram that shows A + B.

• You can pick up and move vectors around, as long as when you put them down they are still the same size pointing in the same direction.
• Vectors that are being added must always touch head to tail.
• You should be able to start at the tail of the first vector and follow the directions of the arrows till you get to the head of the last one.
  • I often suggest people look at it like playing a game of Pac Man (I know, I'm old!) and that Pac Man can only go in the direction the arrows point.

When we follow these rules and draw A + B we should get something that looks like this...

Figure 2

• Notice that you would start at the tail of A, and then move your way up towards B and follow it along to its head. '
• I've also drawn in the resultant (remember it from the last lesson?) that touches tail-to-tail and head-to-head. The resultant really is what A + B are equal to.

Adding B + A

The amazing question now has to be "Will you get the same answer if you add B + A?"

• The quick answer is "Yes!" since you can add vectors in any order you want and still get the same answer.
• In math with regular boring old numbers you can definitely say A + B = B +A... it doesn't matter what order you add numbers in.
  • This is called the commutative property.
• The diagram will look a little different, but the resultant is the same as above in Figure 2. Take a look at Figure 3 below that shows this.

Figure 3

• This one shows B + A because we start with B that points at A and continues to the end.
• Even if we follow from the tail of the resultant (B + A) to its head we still come to the same spot in the end.

This means that you can add the vectors in any order you want. You might measure a different angle than someone else, since your diagram is different and you are going to use different reference points.

• It's like in the last lesson where we saw that [N30°E] is the same as saying [E60°N].

Subtract A - B

This is where things get a bit more interesting.

• What we need to remember here is that in Physics a negative sign simply means "in the opposite direction."
• We can take A - B and simply change it into A + -B
• The negative sign on the B just means that we will need to take the original vector B and point it in exactly the opposite direction (180° from where it's pointing right now.
• Then we will simply add them just like we did in the previous diagrams (touching head-to-tail of course!)

Figure 4

You can see that the resultant we get is different from the one shown in Figure 2 and 3.

• Also be careful the subtraction is not commutative (that just means that A - B ≠ B - A).

Solving Right Angle (90°) Triangles

Most of the triangles you will be dealing with will be right angle triangles.

• If they are, just use your regular trig (SOH CAH TOA) and pythagoras (c² = a² + b²).
• Usually you'll want to be thinking about physics as you set up your diagram (so that you get everything pointing head-to-tail and stuff) and then switch over to doing it like any math trig problem.s

Example 1: A car drives 10km [E] and then 7 km [N]. Determine its displacement.

First, draw a proper diagram:

Notice how this even shows the vectors being added in the correct order according to the question.

• 10 km [E] is shown leading up to 7 km [N]. Start at the tail of the red arrow and follow the path it takes you to.
• If you added them with 7 km [N] and then 10 km [E] you would still get the same final answer, just with a different angle because of a different reference point.

This is certainly a right angle triangle, so just use c² = a² + b² to find out the magnitude (size) of the resultant.

• Try this yourself and see if you get about 12.21 (I'm not being too careful with my sig digs here!)

The angle we should measure is in the bottom left corner.

As a hint, you should probably use tan to figure out this angle. Using either sin or cos will involve using the resultant you just calculated. If you got the resultant wrong, you'd get your angle wrong also.

• Traditionally you measure from the start of the resultant at its tail.
• This also gives us a nice reference line, since we will be able to say how many degrees North of East we are.
• Try to calculate it and see if you get about 35°.

Adding Triangles That Are Not Right Angle

If a triangle is not right angle you have two choices:

1. Use the Law of Cosines or Law of Sines to figure it out.
   This is the "difficult calculations hard if you haven't already done it lots in math" method. If you aren't familiar with them, click here to watch a brief video of me giving you the basics of how to use it.
2. Break it up into horizontal and vertical components, then use basic trig.
   This takes more calculations, but each of the calculations is smaller. More on this after we've looked at the next lesson... for now just keep it in the back of your mind.