https://youtu.be/PE0GwQde2qU
PHYS 1101: Lecture Three, Part Four
Now, we’re ready to talk about this notion of scalar components. It’s the last topic here before I walk you through how to add vectors by their components. The scalar components give us a really quick and useful way of just mathematically jotting down the essence, both the value and the direction of these vector components.
Here’s what I mean. I’ve got a sketch here with an x and a y-axis. My resultant vector is A, and I’ve sketched in here the y component and the x component. These two make up, are equivalent to, the sum of them is equivalent to the hypotenuse here, the resultant.
Let’s focus on one of these vector components. The vector Ay. Of course, for every vector, I need to have a way of clearly indicating the magnitude and the direction. The magnitude of that, we’ve walked through and concluded that I can get a number for that by doing my trigonometry. Off to the side here, I just draw the lengths of the sides of this right triangle. From knowing perhaps the magnitude of the vector A, the hypotenuse, I could go in and figure out the lengths of these two sides using my trigonometric functions.
The direction has to be consistent with this tail to tip direction. Ay has to capture the downward trend of vector A. But notice, because Ay, these component vectors, are always going to be parallel to the y-axis, they can either point only up or down. So mathematically, let’s associate with our magnitude a sign that can indicate for us if this vector point is actually up or down. So, we’re going to do that.
Positive in front of this would mean that this vector component points in the positive y direction. If we put a negative sign in front of it, it’s going to mean that that vector component points in the negative y direction. That’s the case for this vector. Right? It would be pointing down in the negative direction.
So, here’s what we’re going to do. When we started out the class, started out the introduction of vectors, I told you for any vector, we could describe it as having a magnitude of a certain value, let’s say in our case, 38 meters. Then, we have to include the direction information. Here, we could say it points down.
What we’re going to do now is a short hand version of this. Because this is a vector that’s always along the y-axis, I’m going to go with what’s called the scalar component description. That means I am going to write now, for my vector, my scalar component for this vector Ay, I’m going to work just with the scalar component, which is going to be just plain A with a little y subscript. I am going to write that that equals minus 38 meters.
Let me point out what is happening here with each of these components of this description. The first point here is that we call this left side of that equation, the Ay, as meaning the scalar component. The little y subscript that we have here just means this is a scalar component. It’s a part of a vector that’s along the y-axis, or parallel to the y-axis. Then, the sign that’s in front of this number tells us the direction that it points along the y-axis. Negative would be down. Positive up.
What means positive and negative? It’s going to be based on when these axes are initially drawn on your page, when you first pick or draw this initial vector, and you’ve chosen your axes, or they’re given to you in the problem. Just quick short hand. If people only draw an arrow on one end, by default, it’s assumed that that’s the positive y direction. But, it’s better to be more explicit and actually show the positive y. So, that’s what this negative here is going to represent.
Then, the last point to make is the value that you see. Let me highlight that here in blue, the number. That’s what captures our magnitude, the value we got from the trigonometry that we did. So in practice, this is what we’re going to use all the time. We’re going to take a vector, break it up into its components, and then in equations, we’re going to work with scalar components.
It’s going to have this subscript to tell us if we’re doing horizontal components or if we’re working with vertical components. The sign will mean direction, not big or small, but direction of that vector. Then, there’s going to be a magnitude or value. The kind of vector we’re working with will be indicated by the type of unit that’s associated with it.
Let me quick walk you through the mathematical picture of this or justification, because you have all the vector basics now to understand why we do this or why it’s mathematically sound to do this. Rigorously, this justification starts with defining what we’re going to call unit vectors. Small vectors that are going to point along the positive x-axis direction and one that points in the positive y. They have a length of 1.
So, at this first bullet here, let me write for you what we would originally use to describe this vector A. We would write mathematically that the vector A is equal to the sum of the vector Ax plus the vector Ay. With these unit vectors though, let me suggest to you that we can also very rigorously write that this vector A is going to be the scalar multiplication of these unit vectors. Let me write it here and then explain.
I’m going to write this vector A as having what’s called a scalar component Ax times the unit vector x-hat — this is called x-hat, by the way — plus Ay, that scalar component times this unit vector in the y direction. So, we’re just rewriting the x vector component as the product of a scalar component and a unit vector. For the y vector component, we can do the same substitution or the same thing. So, let me pull out that y description for you again. So, that means we’re focusing just on this one part to illustrate Ay. Let me just point out that in order to make this vector A consistent, this component for our particular example would end up being the vector Ay, that component vector, is equal to minus 38 meters times the unit vector ŷ.
Then, let me emphasize this for you. This vector component, Ay, we’re going to write as the scalar component times the unit vector. The scalar component, then, is going to be everything in front of this unit vector y. So, to emphasize, the sign here, that’s what tells us what direction this vector points. Negative tells us it points in the negative y direction. The number here tells us the magnitude of this vector. Then, the ŷ is just consistent with the Ay here, telling us this is a y component vector.
So, the scalar components, that’s what we’re going to work with a lot in this class. I’ve got a bullet here to emphasize that for you. This seems to cause difficulty often. There’s a tendency to want to focus on or to assign the relative size based on this sign. This negative sign here doesn’t mean that this is a small number compared to a big number. All it means is that this is a vector quantity that points in the negative axis direction. What axis? The y-axis. That will always be indicated by the subscript that you see here.
So, this bulleted statement is a good summary for you. We’re going to work with these scalar components. It’ll have a sign and a value. The variable associated with it will tell us if it’s an x scalar component or a y scalar component.
Here’s a lecture quiz question for you to see if you understand how that’s done. Given this vector, what’s the proper sign for the two scalar components for this vector, Ax and Ay?