Introduction to Vectors

Introduction to Vectors

we're going to start looking at vectors. Where we have this geometric shape, whichis a ray, and we use that to display the strength and direction of forces. I do teach pre-calculus. So, we're not going to go real in depth withthese. Not like you would be, say, if you were ina physics class. But, hopefully we'll give you a nice understandingof the basic structure vectors and some of their more fundamental uses. Now a vector is a ray, and it displays thedirection and the amount of force in just the physical world. So, if you were in an area like I live inthat is uh has hurricanes or just some, you know, tropical storms and of course get tornadoesand what not.

But it seems like every time we are on hurricaneseason, our weather channels uh, every you know, any time that a big storm shows up likesto put these uh wind maps on the screen and it's just covered with arrows. Each one of those arrows representing thespeed and direction of the wind which means that each one of those arrows is an exampleof vectors. Another example of vectors is, I've got thisrectangular object here. It weighs supposedly ten pounds. Of course I just made that up. And just say it's a picture frame and we arehanging that picture frame with a couple of wires. What scenario: white, red, green, or uh purple,do you think is the most efficient way of placing those cables to hang this pictureframe? Well that would be vertical. If this entire frame is ten pounds then eachof these vertical vectors in the opposite direction of gravity which pulls down willgive you the most efficient way of hanging that picture, or to put the least amount oftension on those wires.

 Now if I start to let those wires kind offan out and maybe use this orientation the purple lines to hang the ten pound pictureframe then how much force will be placed on those cables? Do you still think that It would be 5 pounds? Uh, you know because it's being shared evenlyfrom the left and right-hand sides. Well, actually, the answer to that questionis gonna be no. We not gonna answer that question in thisvideo, we are just building up the basic structure of vectors and learning all the concepts andthe vocab. But in the next video of applications of vectorswe will learn that question of, well.. if you have a 10 pound iron, you have some vectorsand some kind of given amount of angle. I'm out of colors here. Where we have some sort of fix value of theta… Knowing that is a vertical, uh.. a verticalpull of 5 pounds on this side of the picture with gravity and knowing that's some kindof given angle, what tension is actually on that cable? I'm telling you is gonna be a lot more than5 pounds, the more you let those cables fan out the more you wasting energy in the horizontaldirection and less energy is been placed in the vertical direction to oppose the gravitationalforce.

So, you know i just trying give you a purposeand idea what vectors are formed and how they gonna be used. This kind question of finding forces in differentdirections in what we gonna look at later when you do applications. But for now lets get the basics done. Okay. A vector is a ray, like a stated, it has ainitial point label as x sub 1, y sub 1 and a terminal point label as x sub 2, y sub 2. It has a length which describes the magnitudeof the amount of force that vector is applying when you have this sort of line, double absolutevalue symbol, in vector language that means magnitude. So when you see a vector and when you labela vector simply gonna be usually a single letter and you get this little ray symbol,the same thing you are done in geometry when you first start learning how to… displaylines segments, lines and rays. A vector is a ray so you have the same kind..a little notation a little side arrow above a variable, and that's gonna be a indicationthat's a vector.

So this double line symbol around a vectorname means you want a magnitude or length. you're going to have an angle of rotation,which describes the direction of the vector and, I'm not drawing a initial side, if youbeen watching all my videos, if you been studying trig recently, you know that when we drawangles, we have a initial side and a terminal side. I don't want to draw the initial side, becauseI don't want you to think i'm drawing an angle, I'm drawing a vector. But that direction is going to be the sameas we describe rotations in trig, where standard position rotation where you started the positivex axis and you rotate counterclockwise to indicate a positive direction.

And we do like vectors to be describe in termsof magnitude and direction. Now, if a go back to the example of wind,we don't measure wind standard position rotation, we measure wind as far as like bearings: isit north, is it east, it is a northeast? And you know wanna say the wind is, you know...blowing in the northeast direction at 20 miles a hour, so vectors and, when you are describingit to people, we want to really hear how much force or speed is being applied, and whatdirection is it in. But working with magnitude and direction mathematicallyis not easy at all. So most of this discussion is going to bevectors in terms of x and y components, or horizontal and vertical components. You will see that as we get through here. Our first example of drawing a vector is:I have a vector whose magnitude is equal to two and it's rotation or it's direction is30 degrees.

Well, vector v would look like this. Roughly, of course. Now, this vector has a length of two and it'sdirection is in a direction of 30 degrees. Now here is a little preview, actually thelast example that I am going to do. This vector has a direction of 30 degreesand a magnitude or length of two. How do we describe that in terms of it's horizontaland vertical components? And I get ahead of myself a little bit here,but if you have been studying trig and studying the unit circle quite a bit, 0:06:56.180,0:07:02.260you should actually know right off the top of your head how much is the horizontal componentand how much the vertical component is, because you should understand, or know, the sinesof a 60-30 triangle. That looks like 130, but there is paint onmy board. Sorry about that. So this is a 60-30 triangle: What are thesines?

 What are the ratios of the sines of the 60-30triangle? If it's been so long since you have thoughtof that, maybe you might know what the sine of 30 degrees is. The sine of 30, which is opposite over hypotenuse,is 1 over 2. Okay, that worked. What is the cosine of 30? The cosine of 30 degrees off the unit circleis the square root of 3 over 2. So, either you remember that a 60-30 trianglehas got the sides in the ratios 1, 2, square root of 3, Or if you have maybe forgottenthat, since this angle is off of the unit circle, you know, we came up with it anotherway. But at any rate, for a 60-30 triangle oursides are 1, 2, square root of 3. And I did kind of hint as to what is comingby talking about sine and cosine, so this vector with this magnitude and this direction,can be described in two ways: It could be described as square root of 3i + 1j. And I'll describe later what that little ivector and j vector are. Or my new pre-calculus book doesn't use thesenotations, but my old trigonometry book that I taught out of for quite a few years did. It would describe this vector in terms ofthis: square root of 3, comma, 1.

 And it would use some pointy parentheses,if you will, to describe that horizontal and vertical components. So I did a lot of babbling, trying to explainwhy you want to learn about vectors. Let's pick up the pace a little bit and getthrough all this basic vocab and concepts for you. Two definitions here: equal vector and oppositevectors. Equal vectors have equal lengths and the samedirection. So I've got a couple of examples drawn here:vector v and vector w. They have the same length and the same direction. In other words, they are parallel. We don't normally talk about being parallelwith vectors, but that, you know, appear to be parallel. They are going in the same direction and indeedthey are, but, of course, with vectors, we are normally describing direction with a rotationangle or theta angle. Opposite vectors are again equal in length,but they have the opposite direction. So, I've got two vectors drawn. They are going in the same direction and ifyou were to look at the initial and terminal points and find slope, you would get the sameslope for both of those vectors.

They're parallel. But again, that is not how we describe thedirection of vectors - we describe them with rotation. So while this vector is at an angle of 40degrees, a vector in the opposite direction would mean that you would have an angle measurethat is separated by 180 degrees or pi, because, of course, if I am looking in one directionand I want to go in the opposite direction, I have to rotate 180 degrees. If this is guiding direction of 40 degrees,then an opposite vector would have to have an angle of rotation, or directional angle,that is separated by pi or 180 degrees. So, opposite vectors! Now, I am teaching out of a pre-calculus book,so this is not physics, so we only do vectors for two sections. When I used a proper trigonometry book, therewas a bit more, and I will do more videos, like to approach those problems as well. There was more work done with vectors in theirdesign where you talk about direction and strength, but it really is much, much easierto work with vectors in terms of their x and y components and my new textbook definitelydoes a lot more of that, than my older trigonometry book did, so that's going to be the main ideabehind all of these examples and concepts we talk about through the rest of this video- is vectors in terms of their x and y components. So let's figure that out. I or vector i: horizontal component with avector length of 1, so i is simply horizontal vector with length of 1. J is a vertical component vector with as wella length of 1.

 They are called component vectors, thus theirlength is 1. i is horizontal and j is vertical. So vector v equals 'a' times vector i plus'b' times vector j. I am just going to start saying this as aiplus bj.. vector j. That is equal to x sub 2 minus x sub 1 timesthe i component vector plus y sub 2 minus y sub 1... oops... times the vector componentj. And this idea of naming or coming up witha vector in its horizontal and vertical components is of course if you are given in initial andterminal points.

Which we are going to work out this examplein a second. The magnitude of a vector is equal to thesquare root of X2-X1 squared plus Y2-Y1 squared. Of course this is just the distance formulabecause... well... I am being given an initial point and a terminalpoint. So of course if we want to find magnitudeor length it is the same old formula that you have been using now for many many years. So given this initial and terminal point,that means that this is considered (X1,Y1) and (X2,Y2). So vector v is going to be equal to X2 whichis 5 minus X1 which is -3 with the i vector... plus Y2 which is 11 minus Y1 which is 4 timesthe component vector j. That comes out to be... well any time youhave two negatives next to each other they cancel out, so it is 5+3 which is 8. So 8i... plus 11-4 which is 7j.

 Now this is pretty much how my book stickswith the vector notations, and it really sticks to that i and j notation. My older textbook, my trigonometry book thatI used to teach out of was a lot nicer to simply say that given this initial point andthis terminal point that this vector is going to be labeled <8,7> Now that looks like Iam naming a coordinate. But you will notice the parenthesis are actuallypointy parenthesis. Round parenthesis indicates that you are labellinga point, just a coordinate. The pointy brackets are vector notation andthere is a difference. We are going to be plotting this vector of<8,7> and it will be displayed as a ray.

And when you plot vectors, we are going todo this in a second, but when you plot vectors you can put the initial point anywhere youlike. When you are plotting a coordinate of courseyou always have to start from the origin. What is the magnitude of this vector? Now that is the distance formula. And I have two points given to me. But now that.. you know.. now that I havealready found this condensed name or version of the vector where I have written it in itshorizontal and vertical components, we might want to remember that the distance formulais derived from the Pythagorean Theorem. So while this is the version of magnitudethat my textbook uses in its notes, we might want to remember.. especially when you havea vector already in its component form... that you can simply say whether I look hereor I look here... a times vector i plus b times vector j, that the magnitude of a vectoris also simply the square root of a^2+b^2. That is basically just the Pythagorean...it is the Pythagorean Theorem. That is how we derive the distance formulais from the Pythagorean Theorem.

So the magnitude of vector v is the squareroot of 8 squared plus 7 squared. That is the square root of 64+49. Which is the square root of 113. But I am out of space. And I sure you know how to work the distanceformula. So that is Equal Vectors, Opposite Vectors,and Describing the Vector in terms of its Horizontal and Vertical Components given yourinitial point and your terminal point. Let's do some more. Alrighty then. Let's do a quick sketch and find some magnitude. Vector v is equal to 5i+3j. We are going to sketch and find its magnitude. Now again if your book is not using this notation,you might see that vector v is displayed as <5,3> I like much better for notation, butI got to stick with the notation of my book is using so I don't confuse my students. Vector v <5,3>. Now I could just pick a point, I am goingto do the origin because it is familiar, and go over 5... 1, 2, 3, 4, 5... and up 3. So there is vector v. But with vectors unlikeplotting a coordinate you can put them anywhere. So I could have started let's say here at(2,-3) and go over 5. So 1, 2, 3, 4, 5... and up 3.

Well that is vector v as well. I did not rename so I am not going to callthem equal vectors, though they are. But I am displaying them both as vector v.So with vectors you can start them anywhere you like unlike a coordinate. You know, if I said... probably should have done that first. If I said plot (5,3), then that would simplybe a point on (5,3). It would not be representing a ray, and itwould certainly have to be plotted starting from the origin. Now finding it's magnitude is again the distanceformula. The square root of X2-X1 squared plus Y2-Y1squared. Or remembering that the distance formula justcomes from the Pythagorean Theorem. We can just simply say this is 5 and thisis 3, so the magnitude of v is the square root of (5)^2+(3)^2 which is the square rootof 25+9... or the square root of 34. Scalar multiplication either increases ordecreases the length of a vector.

So if I say I want 2 times vector v. Thenit says if I want to do a scalar multiple of a vector I do k times 'a' and k times 'b'. So I am just going to take 2 times 5, my 'a'value... or I can just do it in this notation... i... plus 2 times 3 times vector j. These are my horizontal and vertical components. And my answer is 10i+6j. And what does that do? Well I start from here again, instead of goingover 5 and up 3 I am going to over 10... so 5,6,7,8,9,10. Then up 6. So 1, 2, 3, 4, 5, 6. And all I am doing, as you would expect, ismaking my vector in this case because my scalar multiple is 2... I am just simply making my vector twice aslong. And we talk about forces certainly wind canpick up and double its speed... I can push harder on something, twice as hardto try and make it start moving. So scalar multiples are just simply increasingor decreasing these natural forces that we have that we are trying to represent withour vectors. BAM! Now in life we generally have more than justone force acting on an object.

So it is very important for us to be ableto find and display resultant vectors. You know, we are going to add and subtractvectors. We are going to combine forces from many directionson maybe an object to get it to move, or remain stationary. So we need to come up with an idea, or a way,of showing this addition or subtraction of vectors both graphically... or just simplywith drawing and working with the horizontal and vertical components. So I have a couple of vectors drawn here. It is 'a' and 'b'. I have only got two. And there are two ways to draw the resultantvector. Like say we are going to add or subtract thesevectors. One way is the parallelogram rule and oneis... I wish I had a name for it. If there is, I don't know what it is. End to End. I like to call it the Triangular Method.

 That is not really very good though becauseif you have more than two vectors you are not going to really draw a triangle. But at any rate, if I want to say draw... Let's draw vector 'a' plus vector 'b'. Well, if I am doing the Parallelogram Rule,that means I am going to start both vectors at the same initial point. So there is a point. Here is, as best I can estimate it, vector'a'. Here is vector 'b' And the resultant is goingto be drawn through the parallelogram that I can make with those two sides. Just the best I can here trying to get a reasonablerepresentation of a parallelogram. Then I am going to draw a diagonal throughthat parallelogram. That diagonal is going to be my resultantvector, or vector 'a' plus vector 'b'. Well, ok. What if I wanted to do vector 'b' minus vector'a'. Well, vector 'b'... Let's go ahead and draw that first from aninitial point. Somewhere around like that.

Vector 'a'. It says minus vector 'a'. Now that means that we are going to go inthe opposite direction. It is an opposite vector. It is negative 'a' instead of positive 'a'So instead of drawing 'a' out in this direction I am going to draw it in the opposite direction,the same length, and let this represent -a... or the opposite of vector 'a'. I think I did forget to do the little negativesign on the Opposite vectors I just talked about a minute ago. Then we are going to draw a parallelogram,draw a diagonal through that parallelogram and that is going to be my resultant. Now a couple of things.

I could be talking about... Maybe I should... do a scalar multiple. But one restriction or one thing I don't likeabout the parallelogram method is if you have more than just two forces working on the same...or working together... if you are trying to find the resultant of more than just two vectors,I find personally using the parallelogram rule rather difficult. How are going to draw a parallelogram whenyou have three or four vectors working together in a system or on an object. If you want to find the resultant of morethan just two vectors, that is going to be a bit of an issue. So what if I wanted to draw the representationof... instead of b-a... vector b minus vector a, I want to do 1/2 vector 'b' minus vector'a'. I want to apply a scalar multiple of 1/2. That is going to take vector 'b' and shrinkit in half. Cut its magnitude, or its length, down bya half.

 And again have a parallelogram and draw yourdiagonal as your resultant. ok, I do prefer though instead of the Parallelogrammethod to draw the vectors End to End. So in other words I want to make... let'ssee... How about 3a+b. Well, 3a+b is going to be... take vector 'a'. There is no negative in front of it. We are going to keep it in the same direction. Just make it 3 times as long. So here is one length of 'a', two lengthsof 'a', and there is the best I can sketch 3 lengths of 'a'. Then instead of drawing 'b' starting fromthe same point I am going to go End to End and start 'b' up here. So vector 'b' is somewhere around that length. Now if I wanted to add a third vector, andsay well there is vector 'c'. Instead of doing just 3a+b, I want to do 3times vector 'a' plus vector 'b' plus vector 'c'. And with this End to End method I can justgo there is 3a, there is b, and here is vector 'c'.

I am going to go back to my initial point,draw straight to my final terminal point, and maybe try and do a straight line... butwhatever. That comes out to be this right here. This purple vector is 3x+b+c. Let's not forget this is a scalar multiple,and this is the resultant vector. So there is my preferred method of drawingvectors in space. That is using... let's call it the End toEnd method as opposed to the Parallelogram Method. Working with horizontal and vertical components. If you want to add vectors or subtract them,all you have to do is add the horizontal components... or subtract them... and then add or subtractthe vertical components. And really it is pretty straight forward. So we have vector v is -2i+7j, vector w is-3i-2j. And you can see those again with those pointybrackets but I am using the 'i' and 'j' notation because my book does.

I really prefer the short hand version ofthis. But, whatever. Find 2W+3v. So we are going to do a scaler multiple onw of 2. So it is going to be 2 times -2i+7j plus 3times vector v... Oops I just got that backwards, didn't I. Let's try that again. 2 times vector w, so it is going to be 2 timesvector w... -3i-2j.. plus 3 times vector v, so 3 times -2i+7j. That comes out to be... and you just distributethrough the parenthesis. So 3 times... or 2 times -3 is -6. 2 times -2 is -4. Then plus positive 3 times -2 is -6i... and3 times -7 gives -21j. And this will have a resultant vector of...let's see. We are going to add the horizontal components,so -6 and -6 is -12i... vector i. And -4j plus 21j is 17j. And so this is how you add or subtract vectorsand do scalar multiples with simple drawings going End to End and this is an example ofcombining scalar multiples and adding and subtracting vectors in terms of their horizontaland vertical components. A couple more screens.

We are almost done. Thank you for watching. I do greatly appreciate it. We do have something called a Zero Vector. It is a vector with a... basically a horizontalcomponent of zero and a vertical component of zero. So it is really just a point as opposed toa ray. So... It does not seem to be like really all thatnecessary. But we will be talking about zero vectorswhen we do applications of vectors and talk about items that are in equilibrium... orare in a static state. That is where all the forces are balancingeach other out so that the object is not in motion at all. Think of say like a picture frame hangingon the wall.

It is stagnate because the wires are balancingout the forces of gravity and keeping that item steady. If you need to take a vector and shrink itslength to 1, and you will need to do that in some of the later formulas that we aregoing to use.... not in this video... but we will need to beable to find Unit Vectors. Finding a unit vector is a vector with a lengthof 1 and the same the direction of a given angle. So in other words, we are just going to takea vector and shrink it. Whatever its length is, shrink it to a lengthof one. And that will be used in later formulas. So this is just vector v divided by its ownmagnitude. You can think of this as a scalar multiple. Instead of dividing by 5 we will be multiplyingby 1/5. So really kind of no big deal. Here is an example. We have a vector with a horizontal component...or a vector of -3i-4j... or <-3,-4> And we want to write a vector that has this samedirection but has a length of 1. So we are going to find the magnitude of thisvector.

The magnitude of v is going to be the squareroot of a^2+b^2. That is going to be the square root of 9+16. That is 25. That is equal to 5. So what we are going to do is... I am going to take this vector and I am goingto divide it by the magnitude that I just found of 5. I write my final answer as -3/5i-4/5j. You might see this same type of notation inthe shorthand form without the i's and j's. One last example... BAM! Now vectors are nice when you want to, ornaming vectors of terms of magnitude and direction is nice, but those numbers are not friendlyto work with. So we want to talk about vectors in termsof direction and strength but we want to work mainly vectors in terms of their x and y components. We want to make sure that we can convert backand forth. We have a little diagram set up here, andlets from that diagram come up with some formulas that we actually already know. Vector v is a times vector i plus b timesvector j. I have got it drawn with its initial sideon the origin just for convenience sake. It does not need to be.

I have got it drawn in quadrant 2, but I amnot going to crunch any numbers, so I am not going to worry about any signs. Let's just look at the cosine of theta. Now the cosine of theta is adjacent over hypotenuse. Since I have put my vector on the origin youcan think of it as x over r. Um... We are going to lose that r idea because vectorsdon't have to be placed on the origin, and thus r... radius of the circle... kind ofloses meaning a little bit. So the cosine of theta is going to be 'a'over the magnitude of v... which is just simply a length remember. Thus 'a', the horizontal... the coefficientof the horizontal component is going to be equal to the magnitude of v, again which isa length, times the cosine of theta.

 You lose... if you kind of just squint andlose all that vector notation stuff, this is just the same cosine is adjacent over hypotenuse,sine is opposite over hypotenuse that you have been using now for... you know.. a coupleof years now... if not at least a few chapters since you have been going through trigonometry. The sine of theta is opposite over hypotenuse,so it is going to be 'b' over the magnitude of vector v which is just the hypotenuse ofthe triangle I have drawn here. That means that 'b' is going to be equal tothe magnitude of v times the sine of theta. And we have a vector again that is in componentform 'a' vector i plus 'b' times component j. Incorporating this idea, that is equalto the magnitude of vector times the cosine of theta plus... with the i, excuse me...plus 'b' which is now going to be the magnitude of vector v times the sine of theta vectorj.

 Let's actually put these into practice. Our example, vector v has a magnitude of 4and a direction of 30 degrees. So we are going to.. let's just sketch thisout. It looks something like this. There is vector v, and there is a directionof 30 degrees. Now, from that magnitude and direction... Now let's go ahead and write that down. The magnitude of v is equal to 4. So from that magnitude and that directionI want to find the horizontal and vertical component. Which means that I need to make a little righttriangle out of that. And, how do I find this little horizontalcomponent of 'a'? I am going to just do this like from trigthat we understand not just really directly use this to be honest.

'a' is going to be equal to... well let'ssee... adjacent, hypotenuse... so cosine of 30 is going to be equal to 'a' over the magnitudeof 4. 'a' is going to be equal to 4 times the cosineof 30. What is the cosine of 30? It is square root of 3 over 2. So this is going to be 4 times the cosineof 30 which is square root of 3 over 2. 4 divided by 2 is 2... 2 square root of 3. Right? The vertical component of b is going to befound through sine. So, the sine of 30 is equal to b over themagnitude of the vector length of 4. So 'b' is going to be equal to 4 times thesine of thirty. The sine of 30 is equal to 1/2. And half of 4 is equal to 2. Now, you might also note that I have drawna 60-30 triangle and if you remember your ratios of a 60-30 they are what help makeup the unit circle after all anyway. Is... opposite 30 is 1 and the hypotenuseis twice as long, and the side opposite 60 is square root of 3. So 1, 2, square root of 3 or x, 2x, x*sqrt(3). So we could have just simply said, from ourknowledge of specialty triangles, that this side is 2 and this side is 2 times the squareroot of 3. And honestly we could have answered this questionwithout even using the trig functions but the point is to show you all of this works. So, that all being said this vector with amagnitude of 4 and a direction of 30 degrees can be describes as 2sqrt(3)i... if you areusing those silly component form things...haha.. plus 2j, or as I would prefer to write ifmy textbook used this notation more <2sqrt(3),2> Ok, last example.

Write the vector v equals -3i-3j in termsof magnitude and direction. So I want to go in the opposite direction. It sounds like it might by new, we are learningvectors but we have been doing this now for like three chapters. Our vector, and I want to again for conveniencesake, put the initial side vertex on the origin. It has a horizontal component of -3, a verticalcomponent of -3. Here is my vector, my vector v. I want todescribe that in terms of magnitude and direction. Well how long have you been seeing me do thisnow? A little right triangle. We have a side of -3, another side of -3. We are going to find the magnitude of thevector by doing basically pythagorean theorem... or exactly. The magnitude of v is equal to the squareroot of -3 squared plus -3 squared, which is the square root of 18, which is the squareroot of 9 times 2, which is 3 times the square root of 2. Now we have the magnitude. What about direction? Man! How long have we been finding standard rotationangles? Again, if you put that for convenience sake,the initial side on the origin... that is really familiar looking. The tangent of theta is... well it was y/x,then with polar form it became b/a, and since 'b' is again our vertical component and 'a'is our horizontal component, the tangent is again going to be b/a, or -3/-3. In this case that is equal to 1. Now you should know that is going to be 225degrees, or 1, 2, 3, 4, 5pi/4. But incase you are still using your calculatorfor this, inverse tangent of 1 is going to give you an answer of 45 degrees.

 That is not the answer but just a usable numberbecause your calculator is just limited on how it can give us... you know... do inversetrig functions. With inverse tangent again, -90 to 90 so thisis really not our answer. We really need 180, because we just passedit, plus 45 degrees. So our final answer from going from horizontaland vertical components into magnitude and direction is going to be vector v has a magnitudeof 9 square root of 2... excuse me... 3 square root of 2 and a direction of 225degrees, or 5pi/4 radians. WHOOOO I am Mr. Tarrou. BAM!

 Go do your homework