**Concept:** Vectors vs. Scalars

Hey, guys. So, in this video I'm going to quickly introduce the idea of vectors, which is a really powerful tool in physics and show some examples of how it works. So, let's jump right into it. So, when you measure things in physics they can be either a scalar or a vector; and the vector is what we want to focus on. It's important to make the distinction between the two. So, scalar is a physical quantity that has magnitude only, and magnitude is just a fancy word for size. So, how big something is, or how long something is. all right, a vector is a quantity that has more information than just that. It has the magnitude, the size and it has the direction of a measurement.

Interaction physics usually means angle. Now, not all physical quantities have direction that's what we have to kind of figure out. So, for example if I tell you I walk 5 meters, I'm telling you the length of my walk, the magnitude of my walk, but I didn't tell you which direction I walked; so, all I'm giving you by saying five meters is the magnitude so that is a scalar. Now, if I say instead that I walked five meters north, now I'm giving you five meters, IÕm giving you the magnitude and IÕm giving north, which is the direction. Because IÕm giving you magnitude and direction this sentence represents a vector. Notice that magnitude by itself is a scalar. So, vector actually has a scalar in it. I like to think of vectors as sort of a box that contain two pieces of information inside of it: a magnitude and a direction. So, to help illustrate that idea a little bit more, we're going to talk about position displacement and distance which are three important variables in motion. Position is basically or simply just where you are and it's represented by the letter X and its measured in meters; and position, the more formal definition, is that it's the displacement from origin; and the origin just means some sort of arbitrary, you make it up, reference point. For example, if you're walking, if you leave your house you might want to say that your house is your reference point, though your origin doesn't have to be your initial position. Speaking of that, so, use your house, let's say you're jealous girlfriend or jealous boyfriend calls and says: Òhey where you atÓ and then you say all the way: ÒI'm just five meters away from the house.Ó The problem with that, is that five meters away from the house could be here or it could be here or it could be here. in fact, it could be in an infinite number, there's an infinite number of possibilities; and if you start plotting these and kind of connecting the dots, you get a circle that would have radius 5. So, your savvy boyfriend or girlfriend realizes that, that's not good enough for him and says: Òwell, which way are you 5 meters in what directionÓ and then you say: Òoh yes, yes, 5 meters South.Ó This is actually where you are. So, you say I am 5 meters south of the house. So, because you are giving a full description of position, requires direction for it to make full sense, position is a vector and here's the idea: whenever you can ask the question Òin what directionÓ that quantity is a vector. So, I ran, I'm 5 meters at 5 meters per second in what way? Did you run to the left? Did you run to the right? So, that's a vector. Let's do some quick examples: your time, one hour has 60 minutes so 60 minutes. Is that to the left? 60 minutes to the right? That makes no sense, if you ask in what direction is those 60 minutes, that would make no sense. This is not a vector it is a scalar. Let's say a certain object is 1 meter long, in what direction? well it doesn't matter, the direction of the object, that doesn't change the length of the object, so, length has to do only with the size; so, it's a scalar. You could orient the object in different directions but that doesn't actually change its length. Mass is, let's say, you are hundred fifty pounds or whatever; and it wouldn't make sense to ask in what direction are you 150 pounds. That's just silly. So, this is a scalar. Temperature is kind of tricky because temperature could be positive and negative; and as you see negatives and positives and physics have to do with direction. However, in the case of temperature, direction, the signs only have to do with whether you are bigger than zero or less than zero obviously, and in this case, zero is just sort of an arbitrary reference point. So, temperature is also a scalar. Now, force is a vector because if you tell me you push a box with the force of 10, it makes a difference if you push it to the left or to the right. So, asking the question Òin what directionÓ would make sense. So, that's the basic difference between the two. LetÕs talk about displacement really quick. So, displacement is defined as change in position. Now, you have to be careful, and this happens a lot in physics, the words displacement and distance in everyday language, are used sort of interchangeably with no noticeable difference.

In physics there's a big difference. You have to be careful about that. So, displacement is defined as the change in position. Now, mathematically, IÕm going to write position final minus position initial and position, remember, we talked about it, it's a vector; so, IÕm going to introduce the vector notation, which is when you put little arrows on top of the letters to represent that they're vectors. So, there should

be an arrow here and IÕm going to explain that this is, IÕm going to write here, that this is a vector notation or vector Symbol, right? That little arrow there. So, another way you can write This instead of X final minus X initial is: you can write X minus X naught. This is the more sophisticated PhD in physics way that a lot of professors will do and X without anything just means final, and X with a zero means initial. Technically, it's not a zero. it's a Naught, X naught, but it doesn't matter. So, these are vectors as well. I can do this. And, any time in physics or any other science or math, they have final minus initial, you can use the Delta notation. it looks like this. Delta X. So, the variable for displacement is Delta X. Position is X, displacement is Delta X. And it is a vector because the direction matters. I could be going from the left to the right or from the right to the left and since it's a vector, Direction matters. another way that you can think of this, is that displacement is the shortest path from start to end and that's sort of a visual representation. I'll show you that very soon. Distance is the scalar, meaning it has no direction of displacement and it's given by the letter D. Now, one way, sometimes this will be the letter S and sometimes they use S for displacement, it's a huge mess, depends on the books, professors, but IÕm going to use D just because distance, D, pretty simple. One other thing about distance is a car's odometer. Now, what the heck is a car's odometer? car's odometer is what tracks or counts your mileage and the idea is that as you're driving, the car doesn't care if you're going to the left, to the right, on north, south, it's always counting; and that's how distance works. And your distance is always going to be either zero or positive. So, your distance must be either greater than zero or positive. Your distance could never be negative, because again, negative is associated with direction and distance is a scalar and it has no direction. So, IÕm going to put here that this is a vector, this is a scalar and distance is D. Let me show you three examples to kind of help illustrate this; one-dimensional motion, two-dimensional motion and circular motion really quick. So, in this case here IÕm going from A, there are two points, A and B, they are 10 meters apart, so, this position IÕm going to call it 0 and this position is 10, in such a way, that the gap between them is 10 meters and that's the case in all of these. A, B, A, B. the only difference is that here I'm going to go from A to B, here I'm going to go from B to A, here I'm going to go from A to B to A; and I want to know what is the distance and displacement for all of these. So, distance, I go from A to B, the gap between them, the separation between them, is 10 meters, so that's my distance. there are texts that are a little more complicated; the definition of Delta X, mathematically, is X final minus X initial. your final position is 10, minus the initial position, you started at zero, so, this is 10. These numbers happen to be the same, that happens a lot, but not always. So, you have to be careful. let's look here, distance, again, you cover 10 meters so that is your distance, doesn't matter that you go into the left, doesn't care about this direction. Delta X is X minus X initial, so, if you plug in numbers your final position is 0 and your initial position is 10, so, if you do this carefully youÕll get a negative 10 and the number itself, 10, is the same, but you got a negative here and this has to do with the direction. Positives and negatives and physics have to do with direction. And in this case, it means that you are going the left, which makes sense according to the diagram. So, notice that the numbers are no longer the same. They're the same here, they were not the same here. They're slightly different. And then, here the distance is, I moved 10 going from A to B; and I move 10 going back from B to a; they're both positive because distance is always positive, is never negative, so this is just 20 meters and my Delta X, my final position, I started at zero, and I ended at zero so, I find at position 0, my initial position 0, my displacement is zero; and look how these numbers are now completely different. So, you have to be careful with the distinction between these two. let me quickly do a two-dimensional motion, I'm going from A, to B, to C. From A to B it's 3 meters and let's say that from B to C it's 4 meters. I want to know the distance and your Delta X, so distance doesn't care about Direction, so it's just 3 and then I go 4; so, my total distance is 7. If you're driving a car, it would show seven irrespectively which way you're going. Delta X is more complicated. Remember the Delta X was, I wrote here, the shortest path from starts to end. So, I'm going to do this. and now, I need to know the length of that blue line. Notice that this forms a little triangle and the length of that blue line is, basically, you'll be found using the Pythagorean theorem: A squared plus B squared equals C squared; and I'm trying to find hypotenuse of that triangle there. Now, this is going to come back later, but in physics, if you have motion in two dimensions like this, we're going to refer to your displacement in the X axes as your Delta X, displacement on your y-axis here as your Delta Y and displacement in two dimensions like this at an angle, as Delta R. we will see more of this later. So, basically what I'm looking for in this case, is my Delta R; and I'm going to do it using Pythagorean theorem. So, it's the square root of the two sides. It's 3 squared plus 4 squared and this is a 3, 4, 5 triangle, so, this is just 5 meters. Notice again how these numbers are very different: seven and five, completely different numbers. And, in the last case circular motion, if you go around the circle from point A back to point A, to go kind of like this, around a circle, I'm going to start with your displacement Delta X. Your displacement is zero because you're back to your initial position, you didn't technically change your position from the very beginning to the very end. You're back to the same position. Now, your displacement is always counting. So, if you are moving, your displays, I'm sorry, your distance is always counting so if you're moving you have to have a distance. And, in this case it's the length of that blue or red line, whatever, and that's the circumference of a circle which is given by: 2 pi r where r is the radius and that's it. And, once again notice how this number is very different from this number. You have to be careful with the distinction between the two and hopefully, this made you realize that a little better. So, that's it for now.

**Concept:** Combining Vectors

Hey, guys. So, now that I've introduced what a vector is, and how it's different from a scalar, in that it has direction, we're going to look into how to do math with vectors, which is something that's going to show up in several chapters in this course. So, you have to be good at it. So, let's jump right into it. Because vectors have direction, during math with them, it's not as straightforward as scalars. So, for example: if you're adding two scalars, 10 kilograms plus 20 kilograms, that is just 30 kilograms. In fact, when we learn in first grade that one plus one is two, that is a scalar addition. They just doesn't call it that because little kids would run. But to add vectors it's a little bit different. I already kind of mentioned this earlier. If you move 3 to the right and 4 up, let me draw this real quick here, if you move 3 to the right and then you move 4 up, your displacement is not three plus four seven, but it's actually five because you're supposed to use the Pythagorean theorem here because displacement is a shortest line from beginning to end. So, it's a little bit different. In fact, because this placement is a vector, we're going to work through a lot of displacement problems and those are the examples I'm going to use to sort of illustrate how vectors work. So, every time I think of vectors think of displacement. And, we're just going to be walking around in different paths and seeing how that stuff works. So, vectors can be added, they can be combined. This is basically vector addition right here. Three and four: I walk three plus four going up. We're adding them and they can be added either graphically, visually, or algebraically, which just means it's just numbers rather than with a graph.

Graphically is more visual obviously, but algebraically could be faster. It's usually faster. We're going to do a lot of graphical stuff first, then I'm going to show you the algebra way of doing it. So, we're going to begin playing with them graphically. And, vectors are represented as arrows. So, vectors are just basically arrows and they're going to go in different directions and it's going to form a bunch of triangles. So, you're just going to be using a lot of tricks to solve these problems. And we're going to add arrows using a method called arrows using a method called tip to tail. The idea is that the tip of one vector will go in the tail of the other. So, if I'm adding these, the tip of the first guy goes on the tail of the second guy and you can kind of keep doing this tip to tail. There's a professor of, that I won't name, but that instead of calling it tip to tail he calls it ass to face. So, I thought that was kind of funny. I guess the ass of the vector on the face of the vector. So, maybe you remember it that way, whatever works. So, example run. let's do an example here: for each of the following drawing or displacement vectors calculate your total displacement. So, you walk 3 to the right and then move 4 to the right. This one is really simple. You walk three to the right, I'm going to call this A=3 and I'm going to tip the tail here, then you walk 4 to the right a little longer, B=4. So, what is your total displacement? Well, if this is A and B, so that C is the total displacement, and this is what C looks like. C is obviously 7. I mentioned how when you're combining these because you walked A,B,C, whatever, its vector addition. So, I can write that C equals A plus B. It's a combination of both paths. Now, these guys here are vectors because there are arrows that have direction, so, I can put a little arrow on top and this is called vector notation. This little arrow right here tells us that this is a vector. Therefore, you have to be careful when you add them. Here, they happen to be sitting in the same axis, so I can just combine them. Since they're both in the same axes I can combine them and I can just do that. I can show you that C is, A is 3, plus B, is 4, so the answer 7. This is the graphical way of doing it; and this is the algebraically way of doing it. Let's do one now. We're going up and down. So, we're going 3 up, I'm going to call this A=3 and then from here we're going to go, make this down like this, 4 is obviously longer so it can be a little longer line, B equals 4. displacement is an arrow from the very beginning to the very end. I'm going to get a different color. The very beginnings here, the very end is here. That is your displacement, that is your C. If I want three up and four down obviously the C has a length of 1. But there's a few details here that I would like to talk about. C is still the combination of A plus B. You always add them up irrespective of the direction you go. Now, what is different is the fact that when you plug the sign, A is going up so, A is positive but B is going down and B is, therefore, negative. 4 negatives and physics represent Direction, that's all they do. So, we're going to put this as a negative and that's because the standard convention is that going up is positive and going to the right is positive. This could change but that's the standard. And we usually represent it by drawing a little diagram like this to show that we're using the direction of positive to be up into the right. So, if you do this you get a -1, which should make sense, this 1 here is the magnitude this 1 here is the magnitude, the size of that vector and this “-“ here is the direction. If going up is positive, this negative tells us that this thing is going down and it is in fact going down. So, I'm going to put a little negative in front of the one right there, I put a negative in front of the 4 as well; and, that's the complete answer. So, that is the easiest way to add vectors, the easiest situation to add vectors, if they're both in the x or both in the y axis. So, vectors in the same axes both X, both Y are just added like you would add any numbers. So, 3 plus 4 and in here, I have to be careful with the sign, just be careful with signs, that, have to do with direction. Now, if they lie on different axes, if one of them is in the x-axis and the one of them is in the x-axis and the other one is on the y-axis, we're going to use trigonometry, “trig”, to add them, to combine these vectors. Let me show you. So, you walk 3 to the right and 4 up. 3 to the right, 4 up. A=3. and then you walk 4 up. B =4. And, your displacement is from here to here, C. To find its magnitude and direction, we're going to use Pythagorean theorem and sohcahtoa. I'll show you that just now. So, the magnitude of vector C, C is a vector so, I can do that, is given by this notation, this is the magnitude of a vector notation, you put the absolute value bars around it. And, it's Pythagorean theorem, so in this case it's just A squared plus B squared. And if I plug in these numbers: 3 squared plus 4 squared, I get 5. 3,4,5 triangle make this easy. And, in this angle here, remember direction is the angle theta. Now, to do this we're going to have to use sohcahtoa, so, we're going to just briefly give you a review of that, I'm going to do a lot more in this stuff a little later, but for now suffice is to show you just this, soh-cah-toa.

Typically, in this situation I already know that this is a 5, but in the beginning, I was only given a 4 and a 3. the 4 is the opposite the 3 is the adjacent. Opposite, adjacent, opposite and adjacent. I'm going to use toa. "Toa" tells us, just as a reminder, "toa" is supposed to remind you that the tangent of theta is opposite over adjacent. That's what it says. But I don't want the tangent of theta, I want theta. So, I'm going to get "theta" out of there. I got to get that tangent out of there, so theta is by itself and the way to do that is to apply the arctangent function to both sides. So ,this is going to be in theta equals the arctangent of opposite / adjacent. Now, I want to take one step further and so, that the opposite is almost always, they are basically always going to be your y-value and your adjacent will be your x-value. So, it's opposite over adjacent but, I want you to remember it as Y over X. So, this is how you find the magnitude of a vector and to find the theta of a vector, it is the arctangent of Y over X. Those are really important, you got to just memorize those and this is the arctangent of the Y value is 4 ,the X value is 3 so you plug this in the calculator, make sure the mode is in degrees, and the answer will be 53 degrees. Cool. so, those are the two answers. This process we call vector composition. The name doesn't matter as much as knowing how to do it but that's what it's called and it's called that because the x and y components are being merged. This is called the x component of the vector and this is called the y component of the vector. I like to think of these as the legs of the vector. So, they're merged to form a two dimensional vector. What I mean by two-dimensional is that it's going at an angle like that. That's actually what I mean there. So, that has an angle, it doesn't sit flat in the X or flat in the Y. It goes on an angle. So, then as for forces, they are also vectors. Forces also have direction. So, if you have multiple forces acting on an object, we can use the same process we just talked about to find what's called the net force. And what is the net force? The net force is an equivalent force on, let's say you have an object that's being pulled by three forces, f1, f2,f3, the net force is one force that could take the place of all of those and I could say that that one force produces the same effect as all of the other forces combined. That's what that is, net force. we're going into some of that stuff later. So, we can use the addition of vectors just like I've shown you to figure out the net Force. So, a box sits in an XY plane, draw an XY plane, here's our little box and you pull on it with a force of six towards the X, so this is you, I'm going to call this A=6, and then your friend pulls on it with a force of 8 towards the Y, so this is B=8. Which way would you expect this force, the net force, to be? or if this box was moving, which way would you imagine at the box would move? you probably imagine the Box moves somewhere in this direction. We can actually figure this out exactly by redrawing this. Okay, I can put one force at a time, I can put A = 6 and then using tip to tail I can keep connecting these forces. B= 8; and this right here, just like it was with your displacement, is your net force. I could have drawn this over here like this middle little box, ensuring that the net force goes this way, these two pictures are the same, the only difference is that I move the 8 from over here to over here to make a little triangle. That's all it is. It looks a little bit more like a triangle, there's a theta there. Now, we're going to find the magnitude and direction. Magnitude of the net force, remember magnitude of a vector, is the vector with absolute value signs around it; and then it's just a square root of its components, these two guys are components. So, 6 squared plus 8 squared, if you do this carefully your calculator get a 10, forces are measured in Newtons so, this is 10 Newtons. The angle is always going to be there against the x-axis I'll talk about that more later, is the arctangent of remember Y over X. The Y value here, is 8 and the x value is 6 so the answer here is 53 as well. All right so, these are the two answers and that's it for this one.

**Concept:** Decomposing Vectors

Hey guys so now that we've talked about how to combine a vector I want to go over the opposite process I want to go over vector didn't decomposition where we are going to get a vector and split it up into it's X and Y components so let's get started so two vectors can only be combined they can only be combine in these two situations here, first if they lie on the same axis they're both X or Y we did this earlier so if they're both on the X axis something like a 3 with a 4 becomes a 7 or something like a, if you think of this as a force a force of 3 pulls up and a force of -4 going down pulls down then this is the same thing as 3 up tip the tail now you get 4 down which gives me the net result the net force of 1 going down, so I can combine vectors that way. I can also combine vectors if one is in the X and one is on the Y axis so something like this I got let's say A equals 3 and then a B equals 4 then I can show you that the C over here is 5 and the magnitude of C is given by Pythagorean theorem and the angle of C is the arctangent of Y over X, in this case B over A but I'm going to write Y over X because that's what I want you guys to remember. Now those two cases are the base places but if the vector is at an angle so the vector is not in the X axis, its not in the Y axis, its at an angle like this it must first be decomposed into it's X and Y components into it's X and Y legs and it looks something like this is the X and Y components that we want. So first I'm going to do this visually without any math and then we're going to do this with math so here each square represents 1 meter so this is 1 meter each so I can count them and see how long these things will be so I want to find the X and Y components of the vector shown so here's the idea imagine that I'm moving from this point to this point over here I can represent this with motion in the X axis and the Y axis instead I can say this motion here the black arrow is the same thing as this plus this, this bottom guy here is called the X component of that vector so if I call this vector A this leg here is A X it's the base of the triangle that I formed and A Y is the height of the triangle that I just formed.

In this case I can the little squares, there's 1,2,3 here so this is 3 it's going to the right remember up into the right is positive that's my standard notation there positive 3 just to be clear and this is 1,2,3,4 positive 4 now I want to find the magnitude and direction and the magnitude of A remember magnitude notation magnitude of a vector has the absolute value notation here it's the square root of A X squared plus A Y squared so these are the components of A and if you combine this you get 5 3,4,5 triangle and then the angle of A which is this guy over here is the arctangent of Y over X in this case A Y over A X. So this is the arctangent of 4 over 3 and that is 53 I'm going to call this guy B and I want to find the X and Y components of B again going down this way now there's two options here I could go like this or I could go like this. We're going to pick the first choice because we want our angle when we make a vector to be against the X axis over here so we're always going to want this angle to be against the X axis so let me erase this and we're going to do this. This is called the this is B X which is the X component of the vector and this is B Y, B X if you count is positive 4, 4 to the right B Y is 3 down so it's -3 and this is B I could put a B here and the magnitude of B is the square root of it's components squared. Doesn't matter the order but I just want to be consistent here X squared and Y squared it doesn't matter that this is negative I could have plugged in as a positive to make it a little faster because it's squared anyway but I just wanted to be more precise there so you see that this is the same number as this alright and the angle theta B here is the arctangent of Y over X the Y value is -3. The X value is positive 4 put this in the calculator very carefully and you're going to get a -37 it should make sense to you that this is a negative angle remember an angle over here is a positive angle and then angles over in this direction are negative angles and we'll be talking about that more later, so negative here means below the X axis cool. The problem is vectors aren't going to fall in those pretty grids that you can just count them. So instead we're going to usually use trigonometry to calculate vector components these X and Y components. Let's do one, You walk 6 meters to the right then you walk 5 meters directed at 37 degrees above the positive X axis let's draw this real quick so you try to make this kind of big so we have a lot of space. I'm going to call this first leg here A equal 6 and then tip to tail I'm just going to connect these things right then I'm going to walk 5 like this and this is B equals 5 and I want to know what is the total displacement? So total displacement is an arrow from the very beginning to the very it looks like this and I'm going to call that arrow C and I can say that because C is the combination of A to B I can say that C is the vector addition of A plus B it's not going to be 11.

So remember what we just talked about if a vectors on an angle it has to be decomposing to its components so this B here has to be changed into B X and B Y let me just make this a different colour here, so this is my B X and this is my B Y. The way I'm going to do this is by using once again soh cah toa and B X is going to be B cosine of theta and B Y is going to be B sine of theta and X is going to go with cosine most of the time, so you can just remember that and Y is going to go with sine. B is the length of B which is 5 cosine of theta the angle here for B is 37 above so it's right there cosine of 37 and if you do this carefully your calculator in degrees you should get that this is positive 4 I am going to put a positive there just to be clear that it's going to the right so this is positive 4 B Y is 5 sine of 37 which is positive 3 right it's going up so it. I found my B now we can figure out C right now that I have all the legs instead of one way you can think about this instead of going 6 and then 5 you can think of it as going 6 then 4 then 3, where did I get these numbers from 6, 4, 3 every problem in physics that has anything going at an angle will be split into X and Y components and that way we make our lives easier and this is your C vector right here so if I want to redraw my C vector it's a triangle like this and this number here this length here is 10 that is my C X and that length here is my C Y which is 3 and if I want to find the magnitude in the angle of C, I can just go back to our Pythagorean theorem and soh cah toa. C is the square root of C X squared plus C Y squared if you plug in all of these numbers 10 and 3 they are both positive if you plug them in very carefully you get 10.4 and the angle of C is the arctangent of Y over X so C Y over C X and again if you plug this in very carefully you get 16.7 degrees and that is the final answer here this process is called vector addition. We're doing this graphically right by drawing your path so one thing I want to point out and we'll talk about this more later is one is that if your angle is one against the X axis or two if it is the absolute angle so if your angle is either against the X axis which looks like that or if it's the absolute angle absolute angle we'll talk about this more later is the angle that is relative to 0 degrees and 0 degrees is right here. So an absolute angle would look like this let me do it real quick, lets say this angle right here is 30 then your absolute angle would be 180 plus 30, 210 those two angles are the same right 30 coming off here is the same as 210 coming off of here wow that was intense. So if you have these two situations you're against the X axis so you have the absolute angle A X will always be A equals sine of theta and A Y will always be A sine of theta and you should absolutely memorise these, these come from soh cah toa

I didn't derive it your professor probably did, your book certainly does and my recommendations rather than deriving every time you just memorise these you are going to be doing this stuff hundreds of times right you don't want to screw it up. So X goes with cosine and Y goes wit sine like I said we'll talk about this a little more later. So remember that forces are also vectors so we can also decompose forces into X and Y components so for example if you have a force that goes on an angle like this this is F and theta I can decompose that F into it's X component F X and its Y component F Y and remember this is the same thing as if you had a box being pulled by a force F X this way and I can draw this F Y to the left over here and it would look like this, these are the same situations. It says here a force of 10 is applied at an angle of 37 degrees so its exactly the same situation here I am just going to draw it again but now with numbers, force of 10, 37 degrees below oh its actually below te positive X axis so below the positive X axis let me draw the X and Y axis here. This is positive axis I wanted it to be below so it looks like this and if this force is 10 and this angle is 37 this is below the X axis so its negative 37 right notice how here they don't tell you that's it's a negative angle you have to figure that out yourself, the unit of force is mu so I want to know what is the X and Y component so again the way to do this I'm going to get this angle out of there because we can't really draw there I'll just do this. Is F X looks like this and F Y looks like this if you want to draw it right so that's decomposing the vectors it makes a little triangle and this is F Y not F X and then F X is F cosine of theta and F Y is F sine of theta I can just plug in the numbers F is 10 cosine of -37, this gives you a positive 8 which makes sense because your going to the right and the calculater knows that and then this is 10 sine of -37 and the calculater gives you a -6 so the calculater knows that this is going down somehow the signs make sense. You always want to get the answer look at your plus the answer into the diagram and see if the numbers make sense if the signs make sense. Anyway that's it for this one lets jump into the next one.

**Concept:** Combining Vectors (Practice Intro)

Hey guys, so now I want you to try solving these three vector questions and I just want to remind you real quick that these are the four equations you need for vector A and you have your component A X here and you have your vertical component A Y and the angle goes right there so you have four variables and these are the four equations that kind of tie all of them together. So give this a shot hopefully you get it right.

**Problem:** You walk 3 m down, then 4 m to the right. Draw your displacement and calculate its magnitude and direction.

**Problem:** You walk 7 m down, then 10 m directed at 53 degrees below the positive x axis. Draw your displacement and calculate its magnitude and direction.

**Problem:** Three forces act on an object: A = 2 N @ 0 degrees, B = 7 N @ -90 degrees, and C N = 5 @ -37 degrees. Calculate the magnitude and direction of the NET force acting on the object.

Consider the following diagram. Which of the following pairs of statements is true?

Ia. **A − B = C**

Ib. **A − D = C**

Ic. **A − C = B**

Id. **B − A = C**

IIa. **A = D − C**

IIb. **A = D − B**

1. Ia, IIb

2. Ic, IIb

3. Id, IIa

4. Id, IIb

5. Ib, IIb

6. Ib, IIa

7. Ic, IIa

8. Ia, IIa

Watch Solution

A vector **A** has a magnitude of 4, and points in a direction of 40° counter-clockwise from the negative x-axis. A vector **B** has a magnitude of 4.5, and points in a direction of 30° clockwise from the x-axis. What is the direction of the vector **2A – 3B**?

Watch Solution

A vector **A** has a magnitude of 4, and points in a direction of 40° counter-clockwise from the negative x-axis. A vector **B** has a magnitude of 4.5, and points in a direction of 30° clockwise from the x-axis. What is the magnitude of the vector **2A – 3B**?

Watch Solution

A vector **A** has a magnitude of 5 and makes an angle of 30° clockwise from the negative x-axis. What are the components of the vector **-A**, using the appropriate signs to denote whether the vector points along the positive or negative axis.

Watch Solution

A vector **A** has a magnitude of 5, and points in a direction of 50° counter-clockwise from the y-axis. A vector **B** has a magnitude of 7, and points in a direction of 30° clockwise from the x-axis. What is the direction of **A - B**?

Watch Solution

A vector **A** has a magnitude of 5, and points in a direction of 50° counter-clockwise from the y-axis. A vector **B** has a magnitude of 7, and points in a direction of 30° clockwise from the x-axis. What is the magnitude of **A - B**?

Watch Solution

A vector **A** has a magnitude of 10, and points in a direction of 50° counter-clockwise from the x-axis. A vector **B** has a magnitude of 8, and points in a direction of 45° clockwise from the x-axis. What is the direction of **A + B**?

Watch Solution

A vector **A** has a magnitude of 10, and points in a direction of 50° counter-clockwise from the x-axis. A vector **B** has a magnitude of 8, and points in a direction of 45° clockwise from the x-axis. What is the magnitude of **A + B**?

Watch Solution

Find the magnitude and direction of the resultant R of the three vectors shown in the figure. The vectors have the following magnitudes; A = 7.9, B = 5.0, and C = 8.0. Express the direction of the vector sum by specifying the angle it makes with the positive x-axis with the counterclockwise angles taken to be positive.

Watch Solution

An object moves 8.0 m north and then 5.0 m south. Find both the distance traveled and the magnitude of the displacement vector.

A) 3.0 m, 13.0 m

B) 3.0 m, 3.0 m

C) 13.0 m, 13.0 m

D) 13.0 m, 3.0 m

Watch Solution

If two vectors have magnitudes A and B, with A ≥ B, and their vector sum has magnitude C, which relation below *must *be true?

a) C ≥ A - B

b) C = A + B

c) C ≥ B

d) C ≠ 0

Watch Solution

A hiker on a trek needs to cross some rough terrain. In order to do so, she has to walk east for half a mile, then head 30^{o} north of east for a quarter mile, then finally she can reach her destination by heading due west for a mile. Because of the indirect path she had to take to reach her final destination, she traveled a pretty far distance. But how far did she actually get from her starting point, as the crow flies?

Watch Solution

Given the vectors in the following graph, write **(A - B) + C** in vector notation ( i.e **a** = a_{x}î + a_{y}ĵ)

Watch Solution