so here's the first lesson for the vectors portion the calculus and vectors course this is lesson 1 insured of vectors you can find all the supporting materials if you follow the link in the description and let's fill up the lesson together in this lesson you're going to understand what a vector is get some definitions and also we'll learn some general notations we're going to use when working with geometric vectors so let's get started [Music] part-1 what is a vector so the first thing you have to understand is the difference between a scalar and a vector a scalar is a quantity that describes the magnitude only and does not include a direction so examples of things that are only magnitudes no direction our temperature distance speed mass those are all just magnitudes what we're going to be focusing on for this portion of the course however are vectors so a vector is a quantity that has a magnitude and a direction magnitude and direction so examples of things that have magnitude and direction while velocity has a magnitude and a direction and forces we're going to be doing a lot of work with forces forces have magnitudes and directions as well so to summarize scalar has magnitude only and vector has magnitude and direction and when working with vectors well there's two types of vectors we're going to be working with geometric vectors and Cartesian vectors in this unit we're going to be focusing on geometric vectors where we actually draw the directed line segments that represent the vector and those directs of line segments they have a very specific length called its magnitude and they also have a very specific direction which we indicate with an arrow head at the end of the vector so like I said in this unit were working with geometric vectors where we draw the vectors that have it's very specific length and direction in future units will work with Cartesian vectors where we're going to define the vectors by using points on a Cartesian grid let's look at a geometric representation of a vector so here is vector a B this vector has a very specific magnitude indicated by the length of this line and it has also a very specific direction indicated by this arrow head here this vector starts here and finishes up e so it's going in that direction so the notation we use to describe this vector a B looks like this we write the starting point of the vector a we write the finishing point of the vector B and then we put a vector symbol above those two letters to indicate what we're talking about is if and this vector symbol always looks exactly like this it's always pointing to the right so the vector we say has two like two parts to it has the tail of the vector and the head or the tip of the vector so tail vector is where it starts in the tip of the vector is where it finishes so this this here is vector a B and sometimes we can name a vector with a single letter so we could name this vector vector V if we wanted to now let's look here the magnitude or size of a vector is designated using absolute value brackets so if we were just interested in talking about not the direction of this but just its length just its magnitude we could indicate that so we're talking about by putting absolute value symbols around the vector that tells us we're just talking about the magnitude of the vector just its length not its direction and magnitude is always a non-negative value which is why we used the absolute value symbol for it now let's look at how we can describe the direction of a vector so when we're communicating the direction of a vector there's a few different ways we could do it the first way we're going to talk about is describing the vector based on the angle moving counterclockwise with respect to a horizontal line so in this diagram here I have vector PQ right I have the vector that starts at P and finishes at Q so that's the notation for vector PQ how can I describe the direction of that vector well in relation to this horizontal line here this vector is 110 degrees counterclockwise from that horizontal line so in describing vector PQ I could say PQ is 14 centimeters right that's its magnitude with a direction of 110 degrees to the horizontal so that's one way we could describe that vector let's try and go the other way what if we have the description and I want you to draw the vector let's draw a vector that is five kilometers that's its magnitude and the direction is 30 degrees to the horizontal so when you hear it describe to the horizontal it's always going to be described in a counterclockwise direction so let me draw that vector for you and when I draw this vector notice that it's the length of the vector is 5 kilometers so we could pick a scale I'm not gonna worry about that too much right now but we could pick a scale saying each centimeter is equal to a kilometer therefore I would make this have a length of 5 centimeters and that would indicate 5 kilometers so there's my vector show it's going in that direction with the arrow head and labeled the angle counterclockwise to the horizontal width 30 degrees and I should also label the length of it as well sorry it was 5 kilometers was the magnitude of that vector so the magnitude 5 kilometers the direction 30 degrees counterclockwise from a horizontal another way we can describe the direction of a vector is with something called a true bearing and a true bearing is a measurement of the angle from north where we start at a North line and move in the clockwise direction if it's a true bearing so a true bearing describes the angle clockwise from north so in this diagram you can see this vector vector U is 135 degrees clockwise from North so if we're going to describe that vector while its magnitude is 2.3 kilometers so 2.3 kilometers let me say at a true bearing of 135 degrees how about if I give you the description of a vector with a true bearing and now you draw it so if we're going to do this it says the vector has a magnitude of 2 kilometers at a true bearing of Oh 60 degrees and a true bearing it's important to know always has three digits so if your true bearing with something with only two digits then you need to describe it by putting a 0 in front so true bearing always three digits so this means 60 degrees clockwise from North so let me go ahead and draw that and I'll keep my scale at each centimeter is a kilometer okay so I've drawn the vector I need to put an arrow indicating the direction of the vector and the angle between the north right North is always up so the angle between north clockwise to that vector was 60 degrees and I should label the magnitude of this vector it was 2 kilometers the third way we can describe the direction of the vector is with a quadrant there so a quadrant bearing is always an angle between 0 and 90 and always you always start by saying whether you're going towards north or south and then how many degrees are you moving towards either west or east so this vector here you would say you're moving toward south and then 35 degrees towards west so we can describe this vector by saying its magnitude is 9.8 Newtons add a quadrant bearing at a quadrant bearing of South right we always say north or south first right we're going towards south and then we say how many degrees towards west or east well it's 35 towards west we say south 35 degrees west that indicates you go south and then move 35 degrees towards west and that's how I would describe vector U for this one let's go the other way where I give you the description of the vector using a quadrant bearing and you draw the geometric vector so this vector has a magnitude of 25 kilometres an hour and a direction of north 80 degrees west that means the direction of this vector would be 80 degrees west of North so here's north here's west and I need a vector that is going 80 degrees west of North so think of it as going in the north direction and then moving 80 degrees towards West so let me just map that off and draw the vector and for this vector I'll make the scale each centimeter is five kilometers an hour so that would mean the length of this line would be five centimeters and draw an arrow indicating the direction of this vector and the angle is north eighty degrees west that means the angle between here and here is eighty degrees and I should also label the magnitude of this vector the magnitude was twenty five kilometers an hour so I should label the magnitude of this twenty five kilometers an hour so in this example we're going to work at converting between quadrant bearings and true bearings this one says write the true bearing of 150 degrees as a quadrant bearing so we have to first start by drawing a vector that has a true bearing of 150 degrees and remember true bearing means angle clockwise from North so let me just map that off quickly so I'm just going to draw a vector that has that angle of rotation from north so what we know so far is that the angle from north clockwise to that vector is a hundred and fifty degrees and we want to describe this as a quadrant bearing law fluttering bearing you describe as either going north or south and then how many degrees towards east or west well this vector here that has a true bearing of 150 you would describe what the quadrant bearing by saying you go south and then how many degrees towards east well there's a hundred and eighty degrees between the North and the South line so the difference between 180 and 150 is thirty so how we would describe this as a quadrant bearing is we would say this vector is going south and then 30 degrees towards East that's how we would describe it as a quadrant bearing Part B says write the quadrant bearing of North 50 degrees west as a true bearing so the quadrant bearing tells us that the direction of the vector is towards north but then 50 degrees towards west so let me just map off that angle and now let me draw a vector that has that that quadrant bearing and so what we know so far is the angle between north and this vector here we know that that angle is 50 degrees right North 50 degrees west means that way and then 50 degrees towards West if we want to describe this as a true bearing which what the question asks us to do we need to know what is the angle between the North line clockwise all the way around that vector so there's 360 degrees in a full circle what we fell 50 degrees short of going 360 therefore the true bearing would be 310 degrees that would be the true bearing that described the direction of that vector so these examples were just to show you that there's multiple ways to describe the direction of a vector often we'll be using quadrant bearings and true bearings so it's good to know the relationship between those two part two let's do some definitions vectors that have the same or opposite direction but not necessarily the same magnitude we would say are parallel vectors right things that are parallel will never cross so parallel vectors have the exact same or the exact opposite direction they don't have to have the same magnitude and hopefully you recognize this symbol here this symbol of two parallel lines means parallel so we need to be able to read this description and understand what it means it says that vector a B is parallel to vector DC so let me just draw those vectors for you so you can see that so here's vector a B and here's vector D see now those vectors are going in opposite directions but they would never cross and therefore they are parallel vectors there's another pair of parallel vectors I want you to notice it says vector a B is parallel to vector C D as well let me draw those two vectors well here's vector a B right starts at a so Tayla a tip-up beat and vector CD has tail at C tip at D now those vectors have the exact same direction different magnitudes but they are parallel vectors that have the same magnitude and direction so they're going in the exact same direction and they have the exact same magnitude we would say that those vectors are equivalent so equivalent vectors have the exact same magnitude in the exact same direction and in the picture below I have three equivalent vectors shown for you now the position of those actors on my screen is different but that doesn't matter the fact that they all are the exact same length and they're all going in the exact same direction indicates to us they are equivalent vectors and let me just show you that with the ruler that they're all the exact same length notice that vector is roughly six centimeters long this vector roughly six centimeters long this vector roughly six centimeters all and the arrows indicate they're all going in the exact same direction so they're all the exact same magnitude and exact same direction so vector a B is equal to vector C D which is equal to vector EF that's what that says oh and each of those vectors I gave a name I call vector a B vector P so we could also say P vector P equals vector q equals vector R but all three of those vectors are equivalent so the position of the vector on the page does not matter all that matters is the length of the vector its magnitude and its direction another definition vectors that have the same magnitude but point in opposite directions so exact same magnitude but point in opposite directions we will call those vectors opposites so opposite vectors have the exact same magnitude but opposite direction so this indicates to us so if we look at these two vectors vector a B and vector B a those two vectors have the exact same magnitude that's of these absolute value symbols mean it means magnitude so the lengths of those two vectors is exactly the same you can tell that here vector a B and vector B a exact same length but those vectors are not equivalent to each other because they're pointing in different directions we can't say they are equivalent vectors but they are pointing in exact opposite directions so we say they are opposite vectors and there's a couple ways we can write that so you can write an expression for an opposite vector by placing a negative sign in front of it or by reversing the order of the letters so if I want to write a vector that is opposite of vector a B so opposite of a B there's two options for opposite of a B I could write vector B a like this that means instead of going from A to B you go from B to a so B to a vector B a is the opposite of vector a B or equivalently to that instead of reversing the order instead of making the tip in the tail reversed we could just multiply vector a B by negative one place a negative in front of vector a B and that tells us we have the same magnitude of s a B but we're moving in the exact opposite direction so these two statements understanding that they mean the exact same thing is very important a negative times vector a B means vector B a for to multiply this vector by negative one it keeps its magnitude but reverses it reverses its direction the last example I want to go through for this lesson and this is probably the most important one now that we have all the definitions and notations out of the way is example two given vector a B draw an equivalent vector vector C D and an opposite vector vector EF so since we know all these definitions now we know in an equivalent vector is we know in an opposite vector is and we know these notations we should be able to do this question and then we'll write an equation to show the relationship between the vectors so for this question you're not going to be able to just copy and paste like I'm going to do on the streets you may need a ruler to measure the actual length of a B because when we draw an equivalent vector we need to make sure it has the exact same length and direction as this vector so I'm just going to copy and paste this vector and I can move this vector anywhere I want on the screen it doesn't change its direction or its length so I'll just place it here and I'll call this vector vector CD because this is the equivalent vector to vector a B so this is vector C D this is vector C D and I'll label this vector here this was vector a B so a B and CD are equivalent vectors because they have the exact same length and the exact same direction now we also need to draw up an opposite vector vector EF so it's going to have the exact same length so here's a vector that has the exact same length but I need its direction to be opposite so let me reverse its direction now instead of starting down here and finishing up here this vector is starting here and finishing down here indicated by that arrow head and it says to call this vector vector EF and I'll change the color of this one so this is vector e F now let's write an equation showing the relationship between all of these so a B and C D are equivalent to each other so I could just write dr. a B equals vector C D now those two aren't equal to EF they're equal to the opposite of EF so if I were to reverse the direction of EF then I would have an equivalent vector so how do i reverse its direction we multiply it by a negative so I can say those are both equal to the opposite of EF and that's what negative vector EF means means same same magnitude opposite direction so this is the statement that we need to write at the end all right that's it for the first lesson for vectors hopefully you have a good understanding of how we draw a vector what all these notations mean what equivalent opposite parallel vectors are and also what quadrant bearings and true bearings are so make sure you try the practice questions and that's it