https://youtu.be/c_YLGO4RH8U
PHYS 1101: Lecture Two, Part Three
Okay, now we’re ready to talk about the next subject in this lecture, which is the idea of a vector. Perhaps some of you have heard about vectors before but I imagine most of you haven’t. There are a lot of physical quantities for which just having a number or a value for makes sense, like the temperature of the room you’re sitting in or the temperature of the drink you’re sipping as you watch this, the brightness of the light that you have in the room. Those are all quantities for which just a single number is sufficient. Maybe it’s 69, maybe it’s 70 degrees Fahrenheit in the room.
This class deals with motion, and with a subject like that there are a lot of quantities for which a number is important but it’s also key to include some directional information with that. Like if a car is driving down the road at 30 miles per hour, where it’s going to be an hour later is significantly impacted by the direction that the car is headed. Is it 30 miles per hour north, South, east or west? So you need this extra information.
And physics and mathematicians have come up with a quantity that we call a vector that is going to cover this for us. A vector is an entity for which you have to associate two pieces of information. You have to have a value and you have to indicate the direction in some way. There’s my bullet for you in the red box to highlight that point.
So an excellent example of a vector that is going to be very important to us in this class would be say, a velocity vector. Velocity, very specifically in this class, will always be a vector. When you say velocity they will always have to assign, or there will be associated with that velocity a value for what the speed is, just literally how fast this object is going. You’ll also have to have some way of indicating, or the problem will give you some direction information. Velocity is always a vector.
Having said that, let’s say a velocity vector might be given to you something of the form — put a colon here maybe — 1 step per second. That’s a unit, although not a SI unit, but a unit of length or distance per second, per time. That’s one piece of information. The other we also need is direction and let’s just make up one here, say due west. It’s key that with this velocity, it’s a vector, so I have to have both kinds of information.
We’re going to run into many different kinds of vectors. The first entity here, let me number that 1 and 2. The first entity, the 1 step per second, is what’s called the “magnitude” of this vector. Note that the magnitude always has units, and those units are going to vary depending on the type of vector that we’re working with. In our case, the units are distance divided by time. We could convert if we knew how long someone’s step is, we could convert from steps to meters and turn this number into so many meters per second. That value with those units would represent the magnitude of this velocity vector.
Part number 2 is the direction. This part is really the same in some ways for all vectors, at least the nature of it is the same. There’s no units associated with it but somehow the literal north, south, east, west, up, down, the direction of this vector has to be indicated. So notation-wise, physicists, scientists use a shorthand notation to indicate a vector and that’s an arrow. Probably makes good sense to you that an arrow and certainly the direction that the arrow points can convey direction. The length can also convey the magnitude or the relative value of it, a bit like a legend on a map.
Let me put here with this arrow that we’re going to schematically use to represent this vector that the length we’re going to use to represent the value or the magnitude. Again, it’s going to be like a legend on a map in that literally the measurement of this on your screen, maybe it’s an inch or half an inch, whatever that length is, that’s going to represent, let’s say, 2 meters per second or whatever physical quantity we’re using this vector to describe. It’s a little tricky here, but I’ll kind of draw an arrow pointing to the end of this arrow that the direction is just conveyed by the literal direction you draw that arrow.
So what you’ll find in a problem is that usually you’re just working on a two-dimensional page and you’ll be comparing, adding, subtracting perhaps, you’ll be comparing some vector quantities and so you may have one vector that has a length and therefore certain value pointed off at this direction and perhaps you have another, let’s say velocity vector, shorter length pointing in a different direction.
You can right away think of the relative comparison here. This is a velocity vector, the speed of this object is smaller than this speed and obviously it’s headed in a downward, perhaps a southeast direction, and this object is going faster and it’s headed in a northeast direction. That’s how you might use this on a piece of paper.
Here I’ve written just a little scale just to give you an example of this legend of a map idea that somewhere, as you first write down this vector, you’ll know that that length represents a certain value.
So take this one for example. If I were to give you this scale and tell you that this distance, this length here represents 5 miles per hour, if this is a velocity vector then what is the magnitude of that vector? What’s the value of it? I would look at this and say, it looks to me like it’s twice as long so it’s going to be 2 times 5 miles per hour, or roughly 10 miles per hour for this vector.
Let me make these a little smaller here for us. Okay. Now that’s a little crunched there. Let me move it down for us so it reads a little bit better.
This is an important bullet having to do with notation. When you read the book, textbooks tend to use the notation of a bold letter to represent a vector. That’s difficult to write out on paper so whenever scientists are by hand writing about a vector they’ll indicate that this variable is a vector by writing some letter to represent it. Like for velocity, v is a natural choice, and then they’ll put a little arrow above it to flag to you that this is going to represent both a value and a direction. I want to caution you that as you’re reading the textbook, you have to look carefully to see if this variable is bold or not.
If the textbook were written properly, in my opinion, they wouldn’t use bold. They would use the arrow over the top like people do when they write it out by hand so it’s clear what you’re dealing with. So keep that in mind. Let me just jot down here for you that this is used in print, say in your book and this is how we’ll write it by hand. In either case we’ll have to make a distinction if a quantity is a vector or not.
The complement to a vector is what’s mathematically called a scalar. A scalar is a number only. It’s a physical entity for which a value only makes sense, like the temperature of the drink, which is now either warmer or colder than it was when I mentioned it earlier. Okay, so memorize that, what the definition of scalar is.
There’s a connection to some terminology here that people use between vectors and scalars. I pointed out that every vector has a magnitude and direction, and you could then say that the magnitude, which is just a number of a vector, that’s also a scalar.
Here’s the convention that people use. If we’re working with some vector, say B, and on paper we write B with an arrow indicating that it has value and direction, whenever we want to talk about just the magnitude of that vector, just the value, usually what you use is the symbol B without the arrow. So plain B will always represent the magnitude of the vector B.
I pointed out that velocity is a common vector that we’re going to use in this class and this is a subtle point a lot of people are initially confused with. We have to be really specific. Velocity will always be a vector. I will always write velocity with a v and an arrow over the top, but we’re going to have occasion to often talk about the value of that velocity vector. That’s the common usage day-to-day that we would call “speed”.
In day-to-day conversations we usually blur this distinction between speed and velocity. We use them interchangeably. In this class we have to be very specific. Speed is only the value, the magnitude of a velocity. It’s a number, so many miles per hour, so many meters per second. When we need to talk about the full velocity of some object we have to accompany the speed information with a direction, due south, due north.
Okay, so here’s your third question for your lecture quiz. Pause it here if you want. Take your time to read that.