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Resultant of Vectors · Acedra
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Resultant of Vectors
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0:00
[MUSIC PLAYING]
0:08
JEFF HANSON: Again we're back. We're going to talk about resultant of vectors, and what we're going to do today is we're going to work that problem. And we're going to use all of the tools we learned last time to see if we can solve it. So the circus clowns have set up the tent at the circus here, and the ringmaster comes out and like, those silly clowns. I don't need two cables there. I need one cable, and so I have to replace two cables with one cable.
0:33
And we're going to call that one cable vector r, the resultant. So in this particular case, vector r is going to be equal to vector p plus vector q. So in our course, this is the first time that we've seen newtons, which is just nothing more than the metric equivalent of force. A newton is a kilogram meter per second squared. And so let's see what we can do here. We know the magnitude of these two vectors. So how do we add two vectors together? Do you remember? Ooh, I know.
1:06
How about our friend the triangle rule? Yeah, the old tip-to-tail rule. So here's that first vector. We'll call that guy p. That's vector p. And then on the end of him is vector q. Here's vector q, tip to tail. And what that does is it makes a very skinny little triangle there, and we'll call this side r. So we've got this skinny little triangle here. What do we know? Well, we know that this side here is 40, and the other side is 60. So this is 40 newtons, and this guy is 60 newtons.
1:52
And so, gosh, I know a side and a side. If only, if only we knew that angle right there, that would be so cool, but we don't know anything. But you know what? Lines are free, so you can draw all you want. So let's draw a few here. So let's do this. Let's put one across there, and let's put one across here. OK, I'm seeing some stuff. How about from horizontal up to vector p? So from horizontal up to vector p, that's 25, isn't it? 25 degrees.
2:28
Ooh, check it out-- two parallel lines and a line between them. So if that's 25, then that is 25, isn't it? So this little guy right here, that little guy is 25 degrees. Hmm. Ooh, how about from horizontal up to vector q? From horizontal up to vector q, ooh, it's both of those angles, 45 degrees. So this guy here is 45. Let's see if that's-- oh, look at this. If that's 45, then this guy in here must be 135, and so that would make this interior angle
3:10
in here 135 plus 25, which that's like 160, isn't it? So this interior angle is 160 degrees. So now what do we know? We know side, angle, and a side. Hey, that sounds like law of cosines kind of stuff to me. Here we go. So here we go. We got r squared equals-- now this time, we don't have a plus b. We got p plus q, so 40 squared plus 60 squared minus 2 times 40 times 60 times the cosine of in between 160 degrees. So let's see. What does r equal? Let's see. Where's my calculator?
4:01
I found my calculator. 40 squared plus 60 squared plus 2-- no, minus 2 times 40 times 60 times the cosine of 160 is equal to 9,710, but then I got to take the square root of that. So second square root answer equals 98.54, so 98.54 newtons. That's how big vector r is. You know what? That's a little vector symbol, but that's not a vector. That's a vector. That's just a magnitude, so I need over here an angle. So where am I going to get the angle? Well, here's vector r, so the angle I need
5:01
is from horizontal. So I need this angle here, whoop. I need that. What should we call this angle right here? Which side goes with this angle? The opposite side is-- this is angle q. So I need angle q, but angle q is just this little bit right here. I need from the floor up. So whatever I find for angle q, I need to add 25 to it, don't I? So how do I find angle q? What do I know that goes together? Well, I know r now is 98.54, and he goes with this. Those two things go together, so here we go.
5:40
98.54 is to the sine of 160. Remember, this is the law of sines. That was the law of cosines. And now I need q, who goes with the 60 over there. So 60 is the sine of q, and so here we go. 60 times the sine of 160 is 20.52, and then divide the 98.54. 98.54 equals-- and then inverse sine of that equals-- so here we go. Angle q is 12.02. We'll just call it 12, 12 degrees. So if angle q is 12 degrees-- that's this little bit in here-- this whole angle must be 12 plus 25, which is-- let's see,
6:46
carry the 9-- 37.
6:53
And that is the single equivalent vector that would replace those two vectors up there. Now, before we get too far, let's look at something. At this point, my students are getting pretty tired of looking at these triangles and doing polar notation because in their back of their minds, they're like, in physics class, I have seen Cartesian, and it's way easier to work with in this. Yes, it is, so let's talk about Cartesian. Let's talk about it.
7:23
I want to talk about I. I'm going to leave this answer here because I want to use that here in a minute. So r is equal to 98.54 newtons at an angle of 37 degrees. I need that here in a minute. So I'm going to erase this, and we'll talk about something completely different for a minute.
7:53
I want to introduce to you Cartesian coordinates.
8:07
Now, you all are very familiar with this, right? The Cartesian coordinate system goes a little something like this.
8:19
We still have our x and y, so in a Cartesian coordinate system, the axes--
8:33
orthogonal. There you go. The axis is orthogonal. What does orthogonal mean? That means that this is at 90 degrees. It only works for 90-degree axis system. If you've got some kind of wacky system, it's not going to work so good. So an orthogonal axis-- it is a right-handed system, which means if you take your right hand and you do three fingers, you're going to get the positive direction. There's positive y. Here comes positive x out at you, and there's positive z.
9:16
Now, some authors like to do this. They like to rotate it. They like to call that-- or let's see. They want to call this x, this y, and call z coming out at you. But it's the same thing. It's just rotated. Now, if you use your left hand, one of the axes is going in the wrong direction. This, your right hand, will give you the positive directions of your angles or of your coordinate axis. So we call it a right-handed system. This system also has unit vectors i hat, j hat, and k hat.
9:54
So here we go. I'm going to do this. This is the way I learned it when I was back at school and the dirt was still shiny. The z direction was here coming out at you. That was the x, and that was the y. Now, if you're looking at a book like Hibbeler, which is the book that we use for statics, Hibbeler likes this system. He likes to call this x. He calls this y, and he calls this z. So don't let that confuse you because you can call an axis anything you want.
10:25
This could be the butterfly axis, the unicorn axis, and the dragon axis. It can be anything you want. So if you see this and this blows your mind because you've seen it in a different configuration, just erase the x and call it what you want, whatever makes you comfortable. It doesn't matter what you call the axis because these x, y, z is just made up things anyway. So a unit vector-- I don't remember what a unit vector is. Oh yeah, it's a vector of length 1, right? So I have this.
10:55
I have this little i hat, and I have a j hat. And then I have, in the z direction, k hat. Now, what does a unit vector do? It doesn't do anything, because I can multiply 1 times anything. 97 times 1 is still 97, right? So multiplying something by a unit vector does nothing except just give it a little direction that says, 97, oh yeah, i hat, which tells me, oh, that's in the x direction, right? So I have these things here. So let's talk about some unit vectors and how to break them-- or not unit
11:32
vectors but Cartesian vectors and how we break them up. So let's do this.
11:41
This will be my vector F, and let's do this. We'll take a little vector flashlight here, and we're going to shine it down on this vector. So this is the y-axis, and this is the x-axis. If I shine that flashlight on this vector, it's going to make a projection or a shadow on the x-axis that looks like this, doo-doo-doo-doo, and that shadow or that projection is what we call Fx. By that same token, if I take another little vector flashlight over here and I shine it this way,
12:21
I'm going to get projected onto the y-axis a shadow that looks like that, and that projection is called Fy. Now, if we do tip-to-tail rule and I moved Fy to the end of here, whoop, look what I have. I have two components, one, two, that triangle rule, go tip to tail, and boom, F is the resultant vector. So that tells me that vector F is equal to Fx i hat plus Fy j hat-- the x component, which is in the x direction, and the y component, which is in the y direction. OK, that's good stuff.
13:04
Now, if I had this angle right here, we'll call it angle theta.
13:11
Let's just write cosine. So for this guy here, cosine of angle theta equals-- what is cosine? Let's see. SOHCAHTOA-- it's adjacent. So adjacent is Fx over-- and this is F up here, right? I'm just going to move that over here. So Fx over f-- which I could rearrange and say, hey, that Fx is equal to F cos theta. How about, same token, let's do sine of theta, which is opposite-- FY-- over hypotenuse, so Fy over F. Again, rearranging this, I get Fy is equal to F sine of theta. So look what I have.
14:04
I have a substitution for Fx and one for Fy that I could put right over there, and so I get this equation. Vector F, some 2D vector, is equal to this F cos theta, F cos theta i hat plus F sine theta j hat. So if I know the magnitude of a vector and I know its angle or its direction, dude, I can write any 2D vector in ijk form, in Cartesian form. So this is a Cartesian vector, not to be confused with a polar notation vector, OK? It's kind of the same information in both of them--
15:03
magnitude, direction, magnitude, direction-- so same information in both vectors, just written a different way. So let's see. Can we use-- now, one more thing. Also, remember this. If I want the magnitude, the how big it is of F, you know what it is. The magnitude of F is equal to Fx squared plus Fy squared, square root, isn't it, because it's just that squared plus that squared square root, right? Pythagorean theorem tells us how big that is. So the magnitude of any 2D vector
15:40
is Fx squared plus Fy y squared, square root. And then we have this equation, which is kind of like a little conversion equation, if you will. Now, let me show you how we can use that. Let's go back to this problem over here. I want to take this vector p, and I want to write vector p, so vector p, in polar notation, where it looks like that. So first, vector p is how big? How big is vector p? Vector p is 40 newtons at an angle of-- what's vector p at an angle of?
16:19
Vector p is it 25 degrees, right? And vector q is equal to 60 newtons at an angle of-- oh, that was 45 degrees, isn't it?
16:37
Now, what if, what if, what if, what if the circus clowns had tied like all kinds of cables on here? You're going to do the triangle rule for that? These two gets a resultant. Then those two get a resultant. Then the resultant of that and the resultant of that gives me the resultant, and then the resultant of that resultant in that vector and another triangle gives me another resultant. No, that would be crazy town. Who would do that? That would be-- we would have to lock you up for sure
17:06
if you were doing that. However, how hard is it going to be to add Cartesian vectors together? Piece of cake, man, because all of the i hats, you can just add them together. Why can you just add the i hats of many, many vectors together? Well, because i hat, i hat, i hat-- they're all going in the same direction, so you can just add them up. j hat, j hat, j hat-- add them all up, right? So Cartesian vectors are super easy to add together.
17:35
You just add all the apples to the apples and then the oranges to the oranges, and you're done. That's easy. So can we take this polar notation of vector and turn it into the same exact vector but turn it into a Cartesian vector? I think we can do it. Watch this. Vector p is equal to-- this is magnitude. That's direction. So where I have that, I'm going to plug it in for F, and where I have 25, I'm going to plug it in for theta. So I get this. 40 cos 25 i hat plus 40 sine 25 j hat.
18:15
There it is, polar vector, Cartesian vector. Let's do the same thing one more time. Vector q is equal to 60 cos 25-- not 25. This time, it's 45. 5 points off for you--
18:34
i hat plus 60 sine 45 j hat. Now, I know that the sine of 45 and the cosine of 45, that's the same thing, and I'm just going to turn this into some numbers here. So p equals 40 times the cosine of 25 is equal to 36.25, so 36.25 i hat plus 40 sine 25 is 16.91 j hat. So there is my vector p. I just plugged that junk in the calculator. Let's plug this junk in the calculator, and what do we get? We get vector q is equal to 60 times 0.707 is 42.42. So how can we get vector r?
19:44
I'm going to add these up here, so here we go. So vector r is equal to-- add the i's together. So that one and that one go together. So plus 36.25 is 78.67 and then 16.91 plus 42.42-- 59.33. Now, this is going to be newtons, and do you believe me when I tell you that that is equal to that? Do you believe? Why do you believe, just because I said it? Let's prove it.
20:46
Let's get rid of all this junk in here, because what is this vector, really? Well, here's what it is. It's 78.67 in the x direction, 78-- that's not an 8-- 0.67. And then it's 59.33 in the y direction, so 59.33. Ooh, look, tip to tail-- how do I find the resultant?
21:22
There he is. How do I find that? Ooh, I wonder how big that is. Well, let me put it in my calculator. 78.67 squared plus 59.33 squared equals-- and then undo square root of the answer equals 98.54. Wait what? What? So r equals 98.54 newtons. Hold on a second. How do I find this angle right here? Well, that can't be right, can it? Tan theta equals opposite 59.33 divided by adjacent 78.67. Let's see what we get. We'd better get 37. Here we go. 59.33 divided by 78.67 equals inverse tan inverse
22:21
answer equals-- do you believe it-- 37, 37 degrees. So here's the deal. We can write vectors. We can add them together, and we know how to do it in the polar form. We know how to write vectors in the Cartesian form and add them together, and as an added bonus, we know how to go from this one back to that one and from that one right back to this one. We can go back and forth between forms there. So what does that make us now? 2D-vector-adding experts. Bring on the problems, OK?
23:01
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