https://youtu.be/ye20b6ByFNc
PHYS 1101: Lecture Three, Part Three
Okay, now we’ve done our reading quiz questions. We’re ready to start our new lecture content. The first topic we need to go over is discussing vector components in a little more detail. You’ve briefly been introduced to them, but let’s work some more examples.
There’s three main points that I’m going to walk you through today. The first one is, you want to be sure, given the axes direction in the problem, that you know how to draw this right triangle so you can then do your trigonometry.
After you’ve got that right triangle drawn and you’ve done your trig to get the proper lengths of the sides, or the values, magnitudes of these components, you want to be sure that you know how to pick the right direction for those component vectors. And then the last point here is the notion of using what we call “scalar components” to represent these vector components.
So let’s start with the first topic here. This is topic one. Let me walk you through some steps to follow to help you draw this right triangle to do your trigonometry correctly.
Let’s start out and say, in a problem you’re given a vector. Either you see it drawn at a certain angle or else it’s described to you as being down from the horizontal, a certain angle, etc. It’s up to you or else it may be given in the problem to pick axes, perpendicular axes that we’re going to use as our mathematical basis to work with this vector to solve the problem. That means you’re going to have to take these axes and use them as our basis for drawing the right triangle here to break the vector down into its components.
Here’s the steps. After you’ve got the vector drawn, you’ve clearly identified or, in some cases, it’s up to you to choose the axes orientation you want. Be sure those are sketched on your page. Then the first critical step to this is I want you to dash lines. Start with one end of the vector, doesn’t matter which one, but just dash a line that’s parallel to one of the axes. Then go to the other end and dash a line that’s parallel to the other axis.
You could have done this either way. If I had started here, I could have a dashed line up here that was parallel to x and then here I could have dashed a line parallel to y. I’d end up with the same right triangle.
Once you’ve drawn that right triangle, then, while you’re looking at that picture, sketch in the two vectors that must, tail to tip, add up to give you this resultant vector. So I always start at the tail and that’s where the tail of the first component’s going to be. Sorry about that. There’s my first vector, tail swinging down to its tip. Then the next vector would start with its tail there and then go to the final tip.
What I’ve sketched here then is the vector By and the vector Bx. The directions are important so be sure that they make sense to you with the tail to tip addition notion. And that’s the point I have down here with my next point too is just check that these follow the tail to tip addition to give you the resultant.
One shorthand way that I use to remember this is that the x-component has to capture the right-leftish trend of this initial vector. See that this vector points down but to the left so Bx has to point and capture the leftward trend of the total vector B. In a similar vein, this vector B has a downward trend to it so By has to point straight down to capture that downward trend. That may help you remember.
So with that help and that introduction, here is another lecture quiz question for you to think about to be sure if you were given or you drew this resultant vector A, that you would make the proper choice for sketching in the directions of the component vectors Ax and Ay. Now, it’s somewhat straightforward or not so bad when you pick the traditional axes, which you will most of the time, with x being horizontal and y being vertical. But sometimes we’re work problems where it’ll make more sense, it’ll be easier for us to use a tilted axes. I’ve repeated the steps here to the right and let’s follow them with this tilted geometry to see that the same idea applies and will give us the right answers.
So, with these axes chosen, my vector drawn, I need to go to one end of this vector. I’m going to go here at the bottom. I’m going to sketch a line here that’s parallel. My eye just first was drawn to the x-axis. I’m going to make this line roughly parallel to the x-axis. I then need to go to the other end of the vector and dash a line that’s parallel to my other axis. This is parallel to the y-axis. This then has become my right triangle.
So, as long as I have some information, I can figure out one of these angles inside this right triangle. Then I can do my trigonometry to find the lengths of these two sides. What would be the directions of these vectors? Well, the y vector, the one that’s parallel to the y-axis, it has to capture a downward trend. It’s going to be in this direction. That’ll be vector By. And then the other side of my triangle has to capture, if I tilt my head so I’m aligned here to the x-axis, I see that in the positive x-axis direction that this vector B does point, has a trend to the right so that direction would be like this.
Quick check here, tail to tip addition, By plus Bx does tail to tip add to give me vector B. So these would be my component vectors and with my angle geometry here I could do my trigonometry to get the values for those.
Now, this next quiz question is a bit tricky. Let me walk you through it a little bit. We’re going to run into this scenario a few times, something that, at first, seems tricky, but, in the end, is really not so bad. Let’s say you’re given a vector like this vector B that you see that’s pointing straight down. What would be the vector components for that? Well, if I try to follow the steps that I’ve outlined, there is no right triangle to draw, per se.
This vector, notice, it points straight down. It is all along the y-axis. It is already parallel to the y-axis. Let me go up here and point out that the vector, when it’s parallel to one of these axes, that is the component. This is the y-component, the one that’s parallel to that axis. The same holds true if you go back up to our previous example. Notice the entire vector here that’s parallel to the x-axis, that is the x-component vector.
What I’m trying to emphasize here for you that is that, because this vector is already entirely parallel to one of the axes, it is the y-component. In other words, this vector is equivalent to By. It has no x-component. So that should help you out with that problem.
Another way to view this is to think of it as a shadow that this vector would cast on the various axes. Let me go to go back up here to the top and show you what I mean. Let’s take this example here that we worked with. So, with this vector, if I were to, let’s say, shine light down to try to illuminate the x-axis, can you see that the shadow that would be cast here on this x-axis is the x-component? Likewise, if you imagine shining light toward the y-axis, the shadow that gets cast would be the y-component of that vector.
So let’s think about that in the context of this vector that’s pointing straight down. For this vector that’s pointing straight down, if I’m shining light toward the x-axis, that means that I’m asking what is the x, the x-component vector. Notice that I’m not going to cast a shadow here. If I turn around then and ask, “Shining light toward the y-axis, what shadow do I cast?”, you’ll see it’s the whole length of the vector. This would then, the shadow on the y-axis is the y-component. That should help you out with that problem.