 Okay. So here is an example problem that I have. Oh, I didn't finish it. Okay. Equal. Let's just give this 1.1. That's a nice remarker. Okay. So suppose that I have some object and at 0.5 seconds it's at location 1, 2, 3 meters. I'm very original. And at time 1.1 seconds it's at negative 2, negative 2, 3 meters. So first, let's do two things. Let's find the average velocity. And second, let's find where it would be at position 3 at t equals 2.1 seconds, if we assume that it keeps up with the same constant velocity. So two parts. So first find the average velocity. So v average is just going to be delta r over delta t. So it's going to be r2 minus r1 over t2 minus t1. You know, I could draw these two position vectors, but I don't have to. I can just deal with them as vectors without having to draw them. So let's go ahead and do this subtraction mode. Remember, change in means final minus initial. So it ends at 2. It started at 1. So if I do r2 minus r1, let me write this as a vector. Negative 2 minus 1 is going to be negative 3. Negative 2 minus 2 is negative 4. And 3 minus 3 is 0. Meters. I'm going to leave the units there. And then t2 minus t1 is t1, t2. It's going to be 0.6 meters seconds. 0.6 seconds. Okay, I should have picked a better number. Let's pick a better number. This is just one. That way I want to use a calculator. Okay, so now I can do this division. I'm dividing a vector by a scalar. So that means I divide each piece. Just checking the mic. Each component by that scalar. So this is going to be equal to negative 3 divided by 0.5. It's going to be negative 6. Negative 4 divided by 0.5. It's going to be negative 8. And then 0. So there's my average velocity. Fun, right? This one won't be too bad. Okay, now if it has that... Changed up to 2. Again, I'm sorry for the boring numbers, but that way I don't have to use a calculator. So now if I want to find where it is at 2 seconds, I can just say R3 equals... Okay, now, remember the position update formula. I could use the change in time from 2 to 3 or 1 to 3. It should give me the same answer. Let me just do R3 equals R2 plus V average delta T. And I was going to write out delta T. It's going to be T3 minus T2. So if I use this position for position 2, I have to use the time for position 2. They have to match. Okay, so let me write this out. R3 is going to be R2, which is going to be negative 2, negative 2, 3 meters. Plus this T3 is 2, this is just 1 second. So when I multiply this by 1 second, I just get negative 6, negative 8, 0 meters. So now I just have to add these two vectors and I get R3. It's going to be negative 2 plus negative 6, negative 8, negative 2 plus negative 8, negative 10, and 3 plus 0, 3 meters. So you should go ahead and try it for yourself. Try doing the same thing, but put an R1 and T1 there. You should get the same value. One other important point. Here, what if I somehow move the origin? So let's say that if I drew this and say here's R1, let me erase this, because you may not be able to see that. Say here's the origin and so here's R1 and there's R2, R1, the vector, R2, and this is delta R. What if I had placed the origin over, let me use a different color, use red. What if I had placed the origin right here and then this would be R1 and this would be R2, but still delta R is the exact same vector. So this is why we notice that delta R doesn't depend on the origin even though R1 and R2 do. So that's an important point. Do we use it later? Okay, that's that.