12 - What are Vectors and Scalars?

Transcript

hello welcome back to physics one the title of this lesson is what are scalars and vectors this is probably one of the first topics that students get to in physics where it's a really a foreign concept most of us have never heard of us what a scaler is before you take physics and most of us probably don't we may have heard of the word vector but we really don't know what it means until you get to physics and so if you don't understand these things ahead of time you get to physics and sometimes it seems really confusing by the end of this lesson you should understand what a scaler is what a vector is how they're different and why they're both important and then in the next few sections we're gonna actually drill down into vectors and scalars even more and start to learn how to work with them but just to kind of like disarm the situation scalars and vectors are very very simple you actually know what they are already you just have never put those words to it before so in in essence we're gonna talk about scalars first then on this side of the board we're gonna talk about vectors now a scalar is a quantity any measurement that only has a magnitude it doesn't have any direction information it's just a magnitude so let's start writing some things down a scalar scale R that's how you write that it's magnitude only now what I wanna be by magnitude only means there's no direction no direction info now once I start giving you some examples of scalars you'll understand why that makes total sense so some examples of scalars would be something like temperature it would be pressure like air pressure it would be time even though we say time flows but if you think about a moment of time there's no direction to it that moment is time equals something so that's a scalar quantity volume of a gas like 34 cubic meters or something like that we've already talked about this one speed it tells you how fast you're going meters per second but it doesn't give any direction information so it's a scalar and just another example mass mass of an object so let's just kind of go through these real quickly and understand what that means so if you have the temperature there's a temperature in space here where my finger is it might be you know 27 degrees Celsius right and there's a temperature over here 34 degrees Celsius those temperatures are different but there's no directionality to this temperature if I'll measure the temperature here which way is the temperature pointing this temperature pointing that way temperature pointing that way no we know it's just not pointing any particular direction we just know that this point in space has a temperature so it's just a number what's what we call magnitude magnitudes how big the number is right all of these share the same kind of thing the pressure you might think pressure has a direction but at this point in space where my finger is the pressure of the gas is acting in all directions pointing in to this point right so at this moment at this point the pressure you know 34 Newton's per square meter or whatever the pressure is it's not pointing any particular direction it's a value at a point okay time we already talked about volume if I have a volume of gas here in this little space we say maybe it's 34 cubic centimeters but the volume it's not pointing any direction it's just a value speed we already talked about in mass this marker has a mass of you know 0.05 kilograms but it's not pointing any direction it's just a value that we give a magnitude so scalars are things you've dealt with all your life they're not pointing any particular direction they're just they're just values right so I've already kind of use my finger to gesture here but just as a picture that you can kind of keep in your mind if you want to think about a three-dimensional space like this right this might be X this might be why this might be Z this isn't this is my kind of a corner of a room showing you here's the flat surface here's the volume of space X Y Z coordinate system right then you might have over here some distance R away from the origin right you might be measuring and we switch colors to make it a little easier to see at this point in space there might be temperature T so you might be measuring it in you know like I said with my finger 50 degrees Celsius or whatever and then if I swing this this R around pointing any other place in space that the tip of this arrow the the the disc arrow is just representing where I'm measuring it I came over here measure here measure here at the tip of this arrow no matter where I pointed is just some number T so it's a scalar quantity the temperatures not pointing in a particular direction all right now because these things are not pointing any particular direction we kind of use regular variable names to describe these guys so scalars are just a number we use regular variables to describe them so in other words we might use T for temperature right T is equal to something we might use lowercase P for pressure we might use V for volume we might use M for mass we might use lowercase T for time or something like this but the reason I'm you might say well this is kind of stupid why is he mentioning this it's because when we have vector quantities vectors include a direction so vectors are going to have a little arrow above the variables to tell you that they're vectors so before we kind of get to the vector part I'm showing you scalar so just know that when you see a variable in a textbook and there's no arrow on top it's just a scalar quantity it's just a number it's some number whatever value it is at that location that's what it is now let's walk ourselves to the other board and talk about the concept of what a vector is I've mentioned it several times in passing in the class because they're so important but now we're finally getting to the actual definition it's a quantity that means quantity that has magnitude just like the scalars of this magnitude but it also has direction so when we give some examples you'll understand immediately why it makes sense that scalars are different than vectors or why we categorize them differently so the best example is gonna be velocity right I'm going 34 meters per second or I'm going negative 37 meters per second the negative sign carries sine information now that is just we talked about so far we just talked about motion along one line where the only way we can go is forward and backward in the side takes care of it right but in three dimensional motion when I can go up and back and and to and from you and so on I can go any direction in space then we'll have vector the vector is the total direction in three-dimensional space that you're that you're traveling so you may not be just going forward and backward you may be going this way that way whatever and so we'll keep track of all of that you'll see how we do it later by keeping track of the motion and x y&z separately but as a quantity when I throw the baseball over there in three dimensions it has a direction associated with it and a magnitude so we call it a vector quantity now another vector quantity one of the most important ones is force when I push on something I'm pushing in a giving direction either I'm pushing along positive X or I'll turn around and I'll push negative X right so the force could have plus or minus components also and that tells you which way you're pushing because it obviously has a direction associated with pushing another quantity is acceleration we haven't talked about acceleration yet we're going to define it and not too long but acceleration you all know from a car is when you speed up right or you might slow down actually we call that deceleration in everyday language when you slow down acceleration in physics we don't really talk about deceleration what we say is you have acceleration in a positive sense meaning that you if you have positive acceleration you're speeding up but if you have a negative acceleration that's what we actually call deceleration and everyday language is when you're slowing down so you see the acceleration variable has vector has a sign associated with a positive acceleration means you're speeding up a negative acceleration not something crazy it just means you're slowing down so it's giving you the direction of your speed up slow down kind of thing so that's why it's a vector there's a direction associated with acceleration not just the magnitude and then we're gonna just list the the remaining here we're gonna get to these much much later but for instance just to kind of give you a flavor magnetic field electric field I'll draw some pictures in a second to help you visualize this but it doesn't matter so much for now because we haven't studied this yet but you've all seen magnets you know magnets interact with each other there's an invisible field we call a magnetic field that field is a vector field in other words every point in space around the magnet has a strength but also Direction it's point to different different ways those are the field lines that come out that we kind of can't see them but that we think they're there we know they're there and we call those vectors vector quantities at every point so this is the difference between scalar and vector so let's draw some quick examples here so if I have a ball that I'm gonna throw and I'll throw it this direction the velocity is 37 meters per second notice that this develop the speed of this ball is a is a magnitude that's the 37 so it has a magnitude check and it also has a direction I'm going up and to the right right so it's a vector quantity that's all you have to ask does it have a magnitude and a direction if yes it's a vector right if we were just talking about speed we would tell you that it was going 37 meters per second but we wouldn't really tell you which way it's going so be it wouldn't be a vector that's why speeds not a vector all right let's take another quick example let's say we have a crate some kind of a box sitting on the floor of my living room or whatever and I'm going to push on it with a force so I'm gonna represent that force as an arrow so force 10 Newtons now I know we haven't talked about the unit of force yet but just a preview for you it's gonna be called a Newton and we'll talk about why it's called the Newton layers after Isaac Newton but anyway it's a unit of force I'm pushing with this magnitude but I'm also pushing at this direction so that's why force is a vector it has a size and it also has a direction associated with it now just to give you a little bit more of a flavor since we're going to be spending so much time talking about vectors we'll just take just a couple seconds to draw a couple of other things let's say you have a instead of a ball let's call it a proton so I'm gonna put a plus charge in the center you all probably in basic chemistry or basic physics so you kind of know more or less what a proton is those are the things in the center of the nucleus well around this proton we we say that this electric field exists and the way that we represent it are these arrows that kind of emanate kind of magically it's not really magic but it comes and emanates from the positive charge and so this thing these these lines here these red lines this is called an electric field now really you can insert a word here it's the electric vector field that's what it really it's a field of vectors in other words at every point in space here there is a value the length of this arrow we're gonna get into that actually in just a second to kind of make it more clear but the length of the arrow represents how strong the field is and the direction of course specifies which way it's pointing so there's this invisible field around all protons right or any kind of charged object that emanates from it and that electric field is what interacts with other matter and pushes it all around that's how we our modern theories of electricity magnetism work and very similar to that you probably have an idea of magnetic field okay magnetic field electric field right now how would we represent a magnetic field we'll just draw our friendly neighborhood bar magnet because I know that you all have some experience with that with playing around with those things and we can't see the magnetic field but we know it exists because it you know we can interact and push on other magnets and such and so the way we'd talk about it in physics is we say there's a invisible magnetic field lines that come out here this is the kind of stuff you learn and you know basic basic science and like third grade but what you didn't learn back then is that this these lines have Direction associated with them that's why we put the arrows these are like a little arrows that exist all throughout space here like this so you see how the magnetic field has a strength and also a direction that's what the arrows are telling you which way is it pointing and the electric field has a strength and a direction and the force and the velocity have a strength in their direction that's why all of them are that quantities and then you have these things that don't have any direction all of these examples they don't have any direction so they're just called scalar quantities all right so how do we represent vectors I told you way back here I said for scalars we use regular variables just like any variable you just write them down okay vectors you might see it written down slightly different ways in different textbooks but most of the time we represent vectors right with an arrow on top sometimes you'll see it written slightly differently but most of the time this is what you'll see if I well I guess I should say the length of the arrow well yeah I'm kind of getting maybe a little bit confusing length of arrow gives you the magnitude in the direction of arrow gives you the direction obviously yeah I kind of got a little head of myself with the way I wrote this down what I'm trying to say by right with an arrow on the top is I'm trying to say in an equation like if I'm gonna write down an equation for velocity or something like this then I'm gonna use the variable V but I'm not gonna write it down like that because if I leave it like that you're gonna think it's a scalar you're gonna think it's some speed or something like this so I write it with a little arrow on top then you know that's a vector quantity okay that's a velocity right vector if I'm gonna write down an equation for acceleration which we will have many many equations will acceleration I'm gonna put a little arrow on top you're gonna know that that's the acceleration right if I'm gonna write down a force which maybe I'll write now Newton's second law of motion or something like that this is a force vector little arrow on top and if just to give you a crazy example that we're not gonna do any time really soon but if I have like e with an arrow on top to go back to our electric field this would be an electric field vector quantity so anytime in an equation when you see a letter with an arrow on top you need to think that's a vector right and when you see an equation with a letter with nothing at all then it's a scalar quantity now I am gonna tell you that some books are different some books instead of an arrow on top just put a bar on top okay the problem with that is that gets confusing because some books indicate a bar on top just a straight bar being the average value of something so like the average of grades in the classroom might be great the variable with the bar on top so it can get confusing if you just put bars on top but when you're doing your homework sometimes you just quickly just put those bars and so that's that's what happens also some books don't put the arrows at all they just bold so you might see V but a capital v with a bold that means vector so it depends on the book you're using but most of the time you're gonna see vectors with an arrow on top now this part of what I'm saying is dealing with what I wrote down here right vectors with an arrow on top this is what I was trying to say here this stuff down here the length of the arrow in the direction of the arrow this is how we write down vectors to represent the values so let's say for instance that have a ball here and I'm gonna draw an arrow like this and I'm gonna say 10 meters per second and the arrow indicates the direction I throw with the ball and the magnitude is indicated by obviously the number now let me go over here and let me throw a different ball but I'm gonna draw it downward and I'm gonna represent this one is 15 meters person now it's important for you to see that the lengths of these arrows are different see this one I'm representing this length of an arrow is 10 and I'm representing this length of the arrow is 15 so with vectors when you write them down as arrows you draw if it's a stronger value a higher magnitude you draw the arrow longer the length of the arrow represents how big the number is the length of the arrow here is shorter the length of the arrow here is bigger because obviously these are different values so just by looking at a just by looking at a picture of arrows you can see that the longer ones are stronger and obviously the direction here is pointing down the direction here is pointing up the directions are totally different right so just to give you another example so this is velocity because we all have experience with velocity let's quickly talk about force because it won't be too long before we actually start dealing with equations that deal with force so just another example what if I have a vector this long I'm representing that as 16 Newton's I told you the unit of forces Newton's we'll get to that later and then I might have another force here acting differently that's going like this and this one might be 30 Newtons do you see I'm not perfect with this but do you see how this vector is about half the length maybe it needs to be a little longer about half the length of this one because this one's 60 and this one's 30 so it's the same kind of thing vectors in general the length of the arrow represents how how strong it is this one's shorter because it's a less force basically all right so reality is what's gonna happen we're going to work here in the first part by representing vectors as arrows as these arrows because graphically it helps you visualize what a vector is the length of the arrows how strong it is the direction of the arrow is which way the thing is acting or moving or whatever it is is doing whatever you're measuring right so we're going to talk about arrows we're gonna talk about graphical ways to write these vectors down in the next few sections but then what's gonna actually happen is we're gonna throw away the graphical pictures entirely and you're not gonna really use those very much solving real physics problems you will use them some but you're not gonna be drawing tons of arrows all over the place to solve problems we're gonna write equations so we use the picture the vector picture the graphical cartoons to visualize it then we're gonna gradually move into kind of getting rid of that and kind of not needing that so much it's very much like learning to add negative numbers we use the number line first to show you how to add them but then after a while we kind of stop using the number line because you don't need it anymore so it is a great tool start with so that's what we're gonna do here so follow me onto the next section we're gonna continue talking about vectors and specifically representing them graphically and how to deal with vectors in physics