https://youtu.be/x14SDinbpps
PHYS 1101: Lecture Two, Part Four
Now when we work with vectors, we’re going to end up working with combinations of them, and we’re going to have to understand how to manipulate them, how to add them, subtract them, what that physically means, etc. This section of the lecture, then, will focus on that.
We’ll start with the addition of vectors and what I’m going to be teaching you here is what’s called the tale to tip graphical approach for adding vectors. I want to introduce it to you in the context of one kind of vector that we’ll run into and use a lot, which is a displacement vector. A displacement vector has a value or a magnitude that physically means as the crow flies, a literal distance, straight line between the start and the finish. That’s the value of it. What units would that have? It would have to have units of length or, say, S.I. units of meters.
But it’s a vector, so what would be the direction of this vector, of this displacement vector? It would have to point from where this object started in a straight line to where this object ended, the finish point. Here on this sketch I’m showing an example of a displacement vector. Let me make it thicker here so you really see where it is. This is where the object started at some location and then it headed due east and it ended or finished at this point. We might use the vector A to represent that displacement vector.
Now consider, a combination of displacements. Now imagine that this object or this person after completing this displacement then turned and walked due north with some displacement represented by vector B. We’re going to consider two scenarios. This one where the person heads due north, compared to the where the person just keeps walking in the same direction. Case 1 here we’re going to compare to case 2.
Case one, after this initial displacement represented by vector A, the person, the object continues heading east and we’re going to represent that with vector B. And now it makes sense to ask, what’s the overall or total displacement? This would physically mean the sum of these two displacements. What’s the resultant? It’s another terminology that we’ll use, the resultant displacement.
From the very beginning to the very end, where did we end up? You can see with this case where the object, person, continued heading in the same direction that the value in meters, the resultant vector, would just literally be the length of the first displacement vector plus the length of the second. It would just be the simple sum of the two magnitudes. We mathematically write that as the resultant vector is the sum of vector A plus B.
But let me point out that just simply adding the scalar value, the magnitude of A and adding that to the magnitude of B to get the literal length of the resultant only works in a special case where the two vectors are in the same direction. Because consider this case, too. This scenario we’re going to start out walking due east. But now for the next step, the person is going to walk due north for vector B. Now, by definition of a displacement vector, the resultant has to be a vector that goes from where the initial start was to the very end.
You’ll notice now that when we talk about this resultant vector R, this displacement vector, which by definition of a displacement vector has to go from the very start to the very end, that it has to be equal to A plus B. But now this length you’ll notice is for sure shorter than the simple sum of this length and this length.
You can’t just directly add the magnitudes of vectors when you need to add vectors together. You just can’t do it. What we have to take advantage of is the geometry of the situation. To graphically do it, what we see is that this resultant vector is what is called the tail to tip sum of these parts, A and B. In other words, the parts that I’m adding together start with one vector. Draw that out. At the tip of that vector now, we put our next vector, the tail, and we draw that vector pointing in its appropriate direction.
Then the sum goes from the initial tail out to the final tip. That is the proper vector sum of these two vectors. That scenario or that description works here as well. In this case they just happened to be lined up, so going from the initial tail to the final tip is the same length as the sum of the two.
This tail to tip approach to adding vectors is going to be true for all vectors. This, just let me write here. This addition approach true for all types of vectors, whatever they physically represent. I like to demonstrate it in terms of displacement vectors because that makes, I think, physical sense to you that the resultant after you’ve done this walk followed by this walk, your total displacement would definitely be off at this angle, the arrow pointing from start to finish, and it would be a shorter length or distance than if you’d walked here and then here.