https://youtu.be/P6Qgy4vr2g4
PHYS 1101: Lecture Two, Part Five
Let’s now go through exercises and show you how, in general, you do this graphical tail to tip addition to find out a sum. Let’s do a couple examples. Let’s start out here with these vectors A and B, and let’s ask ourselves what’s the resultant of the sum of these two vectors. The tail to tip approach would say, I’m going to copy here, this vector, move it over. I start out with A, and then at the tip of A, I’m going to add the vector B by drawing its tail, or lining up its tail, to this tip.
The resultant, then, is going to be the vector that goes from the initial tail to the final tip. This is the resultant vector, which is the sum of A plus B. I have these steps here summarized for you. So, draw the next vector B. Draw its tail starting at the tip of A, and then the sum goes from the initial tail out to the final tip.
Let’s next do an example where we need to add three vectors together. For three vectors, it’s the same idea. We just continue it more times. We’re going to first start out with the initial vector A. Let me ungroup this here real quick. We’re going to start out with A. Let me draw that off to the side. I’ll draw it up here, sure I have enough room. And now, for vector B, I need to add it similar to what I did above with the tail lining up to the tip of A.
But now, I have a third vector C that I need to add to the combination. So, I put C‘s tail. I line it up to the tip now of B. I just keep subsequently adding these vectors together. Regardless of how many I add, I keep adding them tail to tip, tail to tip. Then, the resultant in the end goes from the very first tail out to the very last tip. This is my resultant, my sum of vectors A, B, and C.
It’s interesting that you can add these vectors together in any order. You could have started with B, and then lined A‘s tail to B‘s tip, and then added C. It’s a little bit surprising, but you will get the same answer. So, try that as an exercise if you’d like on your own.
Let me emphasize though, that when you’re working with vectors graphically like this, you do have to be really careful with how you draw them and line them up. You’ve got to be sure and maintain the two critical aspects that you’re representing with each arrow. You’ve got to maintain the length, and you’ve got to maintain the direction very carefully. I had the luxury of just being able to copy and drag these around. So, for sure, the length and the angle was maintained. But when you go to add these up on your own, you got to be careful with that, or you can deceive yourself really quickly about what the right sum would be.
We have another lecture quiz question for you now. Having seen that, go through this exercise yourself. You got these three vectors, A1, A2, and A3. When you do this tail to tip sum, which of these five vector choices here best represents the sum, what you end up with?
Now, for the next section, there’s a few additional facts I want to point out to you about vectors. One of the first is that you’ll run into cases where it makes sense to talk about the negative of a vector. Let me walk you through that in the context of a displacement vector again, because I think it’s physically intuitive.
Let’s take a vector that’s going to represent the displacement of this woman as she climbs this ladder. So, her feet started at the bottom of the ladder and ended here. This is a figure out of your textbook. Make that bigger here. Let me call that the vector D. Now, what would be the physical meaning of a vector –D? Hopefully, it makes sense to you that that would represent a displacement where she now starts at the top and moves down.
That arrow that I’ve drawn, I want you to notice, is going to be the same length exactly. It’s going to be parallel to, you know it’s lined up with the ladder as when she started at the bottom and went up. The only difference is that the arrow points in the opposite direction. That’s all there is to the meeting of a negative vector.
This really comes in handy in the context of subtracting a vector, because subtraction you can think of as the addition of a negative vector. Here’s what I mean. Let’s call it R again, a resultant. What’s the result of starting with the vector B, and let’s subtract the vector A. Let me suggest you think about this as a sum, which we know how to do, of the vector B and the vector –A, the negative of that vector.
Well, the first thing we’d have to do in order to carry out that operation would be to generate ourselves the vector –A. So, I’ve copied it here so we know that its length and direction, the angle’s proper. The only thing I know I need to do is flip the arrow and make it point in the opposite direction. So, here is the vector –A.
Now, let’s carry out the addition as we did above. Let’s take this vector –A, and I’m going to bring over a copy of my vector B. Move that over here. So, starting with B, let me add this vector –A. So, I have to do this new vector –A. I have to draw it at the tip of B. So, the tail of A lines up with the tip of B.
Let me ungroup this real quick, and let me just move the label here over to the side. Because now, when I go in to draw my resultant, it’s always from the initial tail to the final tip. That symbol’s not in the way. This is my answer. This is the resultant, which is B minus A.
So, that covers basics of subtraction. One last fact to point out to you is the notion of scalar multiplication with a vector. That just means scalar, as we defined and learned above, is just a number. When I do scalar multiplication to a vector, I don’t change the direction. I only change the magnitude or the scalar part, the value part of that vector. So, if I initially have my vector A, and I ask what’s 2A, this literally means the value magnitude of A, I have to now increase by a factor of 2. This vector has to become twice as long.
Let me grab a copy of that. Here’s A. I’m going to copy it again and stack it right on top. Try to line that up just right. There we go. Lined it up just right. Let me now, simply, just erase this intermediate arrow. I see I actually didn’t line it up as perfectly as I wanted to. That is the vector 2A. Let me draw it here off to the side. Trying to keep the angle the same, and just the length twice as long. This is the vector 2 times A. If it were 1/2 times A, I would have shrunk it by 1/2, et cetera. Wish I could line that up better. That’s pretty good. Scalar multiplication.
Got another quiz question here for you. Try your hand at this. Choices A through E, which best represents? First, carry out the exercise of multiplying vector a by 2, stretch it to twice its length, and then add the negative of vector B. Let me just write out that hint here for you. Consider this as, first, generate the vector 2 times A, and then add to it the vector –B. So, as we did above, generate this vector –B, and then carry out this tail to tip addition.