https://youtu.be/gW0mZKgDN20
PHYS 1101: Lecture Three, Part Six
Now let’s move on the middle section of this lecture where I want to show you how to add vectors by working with these scalar components. I am going to do that by walking you through one of the simulations that we have access to. It’s for Chapter 1, the only simulation, 1.1. You can get to that by following this link that you see on our class homepage.
So the steps I am going to to walk you through on the simulation are summarized here and let me just drag the simulation over and start walking you through that. I encourage you to pause the movie if you’d like right now, go call up the simulation and to do these exercises for yourself.
The first thing I wanted you to do is to set the, you have two vectors here, a yellow one and a blue one. Go ahead and drag the tip of the yellow one down to 0 and let’s just work with one vector first. This is nice because a feel for what the components do when a vector is in different areas of the graph. So it’s showing you the parallelogram description of these component vectors. You can think of them clearly as the shadow that these vectors, the resultant casts on these two axes. Once way to describe it if I am parallel to an axis, I have no component along the other axis. Anyway, play with that and you can see how these components vary.
So to emphasize that for you I have just done a screen grab, here. And let me past that into our lecture to just jot down for you that what you are seeing there to match what we have done before. I said here is our x and our y axes and this would be, let’s call it the vector B and this is the x component vector, Bx. And this is the y component vector, By.
Now I am going to add the yellow vector back. So I am just going to go back to the origin somewhere and if I click and then drag, I can grab ahold of the tip of the yellow vector and you can see me now being able to slide it around to have a variety of magnitudes and directions.
Let’s put that vector out here somewhere and then I am going to go up here and toggle from the tail to tip mode. Now what it’s done is it’s put the second yellow vector at the tip of our the first one. This is how you would do it if you were going to add this vector, the blue one, to the yellow one.
And so you can see here now how I can swing each of these vectors around independently. Let me pick an orientation for the two of them. Then up here I am going to click on the “show sum”. You can see that it drew the vector as we expected. The resultant goes from the initial tail out to the final tip. It’s the sum.
All right here is the key that I wanted to point out to show you the strategy behind this, adding vectors by components. Let me take the sum off for just a second. So recall these are the two scalar components for the first vector. With the yellow vector drawn at the tip here, you can see the shadow cast on the two axes by this vector. So these would represent it’s two scalar components for yellow.
When I show the sum, if you focus just on the red vector now, you see that its scalar components are these small red lines, the thinner red lines. These are the shadows that red the red vector cast on the two axes. If you look closely, though, I want to point out, you notice that the length of this red scalar component really, literally, is the sum of the blue and the yellow. Because these components are all along the x-axis, the sum of those has to give us all of the x component or the x part of the sum, the red vector. The same thing with y. Because the blue y component is along the y-axis as is this yellow y component, I literally just have to add these two parallel vectors together to get the sum along the y component, which is the shadow that our resultant vector casts on the y-axis.
Okay, so I am going to analyze that a bit more for you and show you how to set up a nice table, which will look a lot like this in the end, to do this for yourself. But while you’re here, go ahead and play with changing these vectors around a bit. Let me point out, as an example, if I swing this vector around so that this blue vector has a negative component along y, the yellow vector here, it has a positive component along y. So those two components would have opposite signs. The blue would be negative and the yellow would be positive.
When I add that positive and negative number together I end up with the red, which in this case, is slightly negative. And that is the appropriate shadow that this red vector casts. The resultant does have a y component that’s small and negative, it’s pointing down. There’s a downward trend to this vector. So play with that and focus on those two components and see how that makes sense.
Let’s go back to something, though, that looks like this, where all these vectors are up in this first quadrant and let me show you a table approach to working this out. That’s summarized down here for us. So here I have done a screen grab for these two vectors for an orientation similar to what I have show you above there.
Here are the steps you want to follow. We are going to end up making this table but first, when you look at these vectors, don’t look at the red one first but the blue. For each of these vectors the blue, you are going to have to find the x and the y components. For the yellow vector, you need to find the x and the y components.
Make yourself a table. Make one column titled x, one column titled y, and then what we are going to fill in along these rows, are the blue components, the x and the y. Next will be the yellow components or the gold for x and y. And then what we end up with is just adding down these columns to get the vector, the scalar vector sum.
So down the x column, I’ve got blue’s x scalar component, the yellow’s scalar component. That number plus that number adds up to the red scalar component for x. In the y I got the same thing, the y scalar components. I add these two parts to get the y scalar component for the sum, that red vector.
An important point, I’ve got it emphasized here for you, what has to go in here are the literal scalar components. Meaning there has to be the proper sign here to indicate the direction of these vectors. You need that proper sign to combine to give you the correct sum.
So here’s some steps to follow. The first thing you need to do when you’re adding vectors by their components, is that you want to find these scalar components for all the vectors that you are adding. Usually it’s going to be two, might be three. For each vector take your time, do your trig, find the scalar components, be sure the signs are correct.
Then generate a table, similar to what I have shown here. If you are adding just two vectors you are going to have two rows. If you are adding three, you might have a third row here. Once you’ve generated that table, you just need to fill in the numbers. Fill in all of the x components, fill in the y components and then go down the columns and add them up. What you end up with is the resultants. The Rx is going to be the sum of this column. It could be positive or it could be negative and then you are going to have a certain Ry. The positive or negative, the sum of these numbers to give you that.
Okay, then once you got that, usually the problem is going to ask you to go a bit further and determine what the magnitude of this resultant vector, or it might ask you what the angle is of the resultant. Meaning, the focus is on this resultant, R. What’s the magnitude? What’s the length of that and what’s the direction? Maybe they want an angle.
If you wanted the magnitude, because you will have the two sides of this right triangle, I would use the Pythagorean theorem to get the hypotenuse. Let me sketch, here, for you, for the example that we show here, this resultant vector here is often at an angle. It’s pointing up and to the right, that’s R. When you add down, these columns and you get these numbers for Rx and Ry, they should be consistent with the two component vectors looking like this. This would be Ry and Rx.
This means that Ry should end up being a positive number because that component should point up. Rx should also be positive because it points to the right. No matter what direction they point, you need to do the sum of the squares to get the hypotenuse. This side squared plus this side squared, square root.
Now, let’s say they ask you for an angle, what would you do? Again, it’s an exercise of trigonometry, but this time we would use a sine or cosine function. They may tell you what angle they want or it could be up to you to specify. Let’s say we want to go after this angle. We would want to, then, pick our trigonometry that would relate this angle to some of the sides that we know.
If we have already figured out the magnitude then we could use any trig function we wanted. We could use tangent, which would be the ratio of this side to that side. Cosine of this angle would be equal to the ratio of this side to the hypotenuse, etc.
The point I have here, though, about finding the angle is when you are doing that, don’t worry about the signs of these two scalar components. Just do your trigonometry with the positive values. Go back to thinking of this as just a right triangle and you are trying to find this angle,