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Anderson Video - Displacement Vectors and Adding Vectors

Professor Anderson
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>> So what do we mean by displacement? Displacement is moving from one location to another. So if you think about moving in a 2-coordinate system, x and y, a 2-dimensional system, we need to know a couple things. We need to know where you started and where you ended. So let's say you started there and you ended right there. How would I define those positions in this coordinate space? The vector from the origin to your initial location is s sub i. The vector from the origin to your final location is s sub f. And therefore, delta s is going to be the difference between the two. It's going to be the final location minus the initial location. And we know exactly what that should look like. It should look like that right there. From where you started to where you ended, that's what delta s looks like. Now, let's convince ourselves that this is true based on our rules for vector addition. So, one thing we said was, if you are subtracting a vector it's exactly the same as adding the negative of that vector. Okay. s, f looks like this, up and to the right. s, i was up and to the left, which means that the negative of s,i -- you just flip the arrowhead, okay. Same length, just opposite direction, so it would be down and to the right. So it would be something like that. And now I'm just adding two vectors. How do I add two vectors? I start at the beginning, I go to the arrowhead on the end of the last one. And now that delta s, which is roughly how we drew it over there, is that long and it's in that direction. Okay. So, adding vectors means this tip to tail method applies and any time you subtract it you're really just adding the negative of it. Let's talk about adding vectors when we have more than just two vectors. And let's say we have a whole bunch of them. Let's say we have vector a that looks like that, vector b that looks like that, vector c that looks like that, and vector d that looks like that. Okay. They each have a length, that's the magnitude of the vector and they each have a direction indicated by the arrowhead. Alright. How do we add these things up? Well, let's write down the resultant vector r, which is the sum of those things. r is equal to a, plus b, plus c, plus d. And now let's think about how to do this graphically. Alright. What we said was, when we're adding two vectors we just draw the first vector somewhere. There's my first vector a, roughly the same length and the same orientation. And then I take the tail of the second vector and add it to the tip of the first vector. So there's the tail of b attached to the tip of a and it's heading off to the right. And now we take the third vector and we move it into position such that it starts at the arrowhead and moves off in its particular direction, roughly like that. And then we do the same with vector d. It starts at the end of c and goes up in that direction. Okay. So what does r look like? r, you start at the very beginning of the problem and you go to the very end of the last arrowhead. So r is that little arrow right in there, and that graphically will now allow you to approximate what the length of that vector is going to be and what angle it's going to be. And so you can double-check with your calculations when you're all done.