Displacement is the change in a position vector. It is not the same as distance, which is a scalar measurement. The net vector of multiple displacement vectors if found according to the rules of vector addition.
So let's talk about displacement. What is displacement? Displacement is the change in your position vector. It is not the same as distance; there is the biggest mistake that I've seen my students make in the past is that they think that displacement and distance are the same thing. They're different, displacement is a vector and what that means is that it's allowed to cancel. So for example if I start right here and then I move over here I have displaced myself about a half a meter to the right as far as I'm concerned probably to your left. So displacement has a direction associated with it, you can't just say your displacement was 3 meters. Your displacement was 3 meters to the right and it's important to represent that because if I move a meter to the right is not the same thing as me moving a meter to the left, same distance not the same thing.
Alright displacement can also be allowed to cancel, so if I start here and then I displace myself, then I displace myself again my displacement now is zero. The distance that I traveled was not zero but the displacement is. Alright, so displacement is a vector if I start here and then I go here, my initial position was this one, here's the origin, the initial position vector is this one to 0.1, the final position vector is this one to point 2. So my displacement vector is final minus initial so it's this vector right here, delta r, r2-r1 okay notice that it's a vector and it's exactly what we mean by displacement, you start at 1 you go to 2 so what was your displacement? It was that so itÃ¢Â€Â™s really really really easy and obvious but you just have to pay attention to what it means.
Alright let's do a more complicated example suppose that I start here at position 1 I go to 2, then I go to 3, then I go to 4, then I go to 5 alright each displacement goes like this 1 to 2, 2 to 3, 3 to 4, 4 to 5. Now if I want the net displacement it will be this displacement plus this one, plus this one, plus this one, plus that one. Now of course you know what it has to end up being because what is the displacement from 1 to 5? ThatÃ¢Â€Â™s it 1 to 5 and that is what you'll get if you add all those vectors together. It'll just be more annoying to do that then to just draw that vector that goes from 1 to 5. The reason why it's so easy, is that displacement is allowed to cancel, so it actually makes displacement much easier to calculate in general, especially in complicated problems than distance.
Notice if I wanted to know that distance that I traveled, I would have to find the length of that vector, the length of that vector, the length of that vector and the length of that vector and then add them all together and I don't want to do that. I want to just draw a red vector from 1 to 5 and thatÃ¢Â€Â™s why displacement is easier. Alright now let's just do a real simple example in one dimension, I donÃ¢Â€Â™t have to really worry about vectors because a negative sign will carry all the information about direction that I need. So displacement equals x final minus x initial. Let's say that I started at 5 meters and I ended up at negative 3 meters and I wanted to know what's the displacement well final minus initial negative 3 minus 5 so the answer is negative 8 meters or sometimes you might say it as 8 meters to the left and thatÃ¢Â€Â™s displacement.