Subtracting Vectors

Subtracting Vectors

Let's go back to the problem of the heavy box that you and your friend are trying to move. If you didn't communicate properly first, you both might think that you should pull in your own directions! Imagine you stand behind the box and pull it towards you with a force \(\vec{{F}_{1}}\) and your friend stands at the front of the box and pulls it towards them with a force \(\vec{{F}_{2}}\). In this case the two forces are in opposite directions. If we define the direction your friend is pulling in as positive then the force you are exerting must be negative since it is in the opposite direction. We can write the total force exerted on the box as the sum of the individual forces:

What you have done here is actually to subtract two vectors! This is the same as adding two vectors which have opposite directions.

As we did before, we can illustrate vector subtraction nicely using displacement vectors. If you take \(\text{5}\) steps forward and then subtract \(\text{3}\) steps forward you are left with only two steps forward:

What did you physically do to subtract \(\text{3}\) steps? You originally took \(\text{5}\) steps forward but then you took \(\text{3}\) steps backward to land up back with only \(\text{2}\) steps forward. That backward displacement is represented by an arrow pointing to the left (backwards) with length \(\text{3}\). The net result of adding these two vectors is \(\text{2}\) steps forward:

Thus, subtracting a vector from another is the same as adding a vector in the opposite direction (i.e. subtracting \(\text{3}\) steps forwards is the same as adding \(\text{3}\) steps backwards).