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Unit vector notation (part 2)

More on unit vector notation. Showing that adding the x and y components of two vectors is equivalent to adding the vectors visually using the head-to-tail method. Created by Sal Khan.

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  • starky ultimate style avatar for user Andrew
    Why do vectors have to be arranged "head to tales" in order for the addition to work out?
    (35 votes)
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  • leaf green style avatar for user R
    We have been taught vector additions differently, do you know how to use the parallelograms of forces?? We had a question that asked "if a 5.0N force acts East and a 12.0N force acts North due West at an angle of 60 degrees what is the resultant vector and the angle?" It requires the use of parallelograms of forces and I am not sure how to do it. PLease help.
    (9 votes)
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  • male robot donald style avatar for user Dwij Hirpara
    If we wanna add two vectors, we arrange them HEAD to TAIL right? then what if i wanna subtract them? how do you arrange vectors graphically when you subtract them?
    (4 votes)
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    • leaf grey style avatar for user Nabla ∇
      The vector you want to substract out must change to its opposite direction and then arrange graphically using the head-to-tail method.

      If you have the vector (3, 2) and you want to substract it from other one, you change its direction to (-3, -2).
      (3 votes)
  • leaf blue style avatar for user Ahmed
    In the graph, to get vectors a and b we rely on x and y axes which guarantee forming a right angle even if we need to do shifting. however, after getting vectors a and b (green and red arrows) at what guarantees that they will be perpendicular on each other and form a right angle?
    for example: if vector b equation was b=2i+2j instead of b=2i+4j it will form an obtuse angle with vector a instead of right angle and trig functions will not work. what do we do in that case. and will the analytical way still valid with such equations?
    (4 votes)
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    • piceratops ultimate style avatar for user Timm Wellman
      This is one of the biggest reasons that the unit vector notation exists. With it you don't have to have a right triangle or use obscure trigonometric functions.

      Also, in the case of he video, the vectors aren't at right angles, even though they seem that way visually. If we multiply the directionless slope of the two vectors together and get anything but -1, then they are not perpendicular.

      slope of vector a=2/-3 or -2/3
      slope of vector b=4/2 or 2

      since -2/3*2=/=-1 then the two vectors aren't perpendicular.
      (1 vote)
  • male robot hal style avatar for user Sufyan Sajid
    can 2 vectors be added if one is in 2 dimension & the other one is in 3 Dimension ?
    (2 votes)
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  • piceratops ultimate style avatar for user Mehrab Jamee
    how would you subtract using the head-to-tail method?
    (4 votes)
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  • male robot donald style avatar for user snehit202
    is the triangle vector sum related to the length? because they say the sum of 2 vectors is equal to the 3rd by triangle rule but a triangle is not formed if sum of 2 sides is equal to the 3rd one.
    (3 votes)
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  • leafers sapling style avatar for user amylkearl
    Does anybody know what a vector diagram is? I don't understand
    (2 votes)
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  • male robot donald style avatar for user Trypus
    SAL! You are someone to be believed, if not blindly then maybe after some thought. You are helping so many people's lives! (this happens at )
    (2 votes)
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  • starky ultimate style avatar for user mattchuuu904
    Why do vectors have to be arranged ''head to tales in order for addition to work out?
    (2 votes)
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    • male robot donald style avatar for user Venkata
      When drawing an arrow to represent the vectors, the tail stands for the start point and the head stands for the end point. Now, if you join vectors head to head, it would mean that you traveled some distance to a point x, miraculously teleported somewhere else to some point y and traveled back to point x.

      That is why we join head to tail and not head to head, so that we don't breach the laws of time and it makes sense to add up the vectors!

      Hope this helped.
      (1 vote)

Video transcript

Welcome back. In the last video, I at the end of the video, like I always do in the attempt to confuse you, I told you that if I had two vectors-- And let me just make up some new ones, so I can draw them visually in a second or two. Let's call the first vector a. Let me do a different color. This toothpaste color is getting monotonous. Let me do something that looks relaxing. Let's call a first vector a and, I don't know, let's make it interesting, let me say it's minus 3 times the unit vector i plus 2 times the unit vector j. And then I have vector b. And that is equal to, 2i, so two times the unit vector i. Plus, 4 times the unit vector j. In the last video I said, well, the whole reason why this unit vector notation is even -- Well, one of the reasons, we'll see that there many reasons why it's useful. One of the really cool things about it is, before when we added vectors, we would put them head to tails, and then draw it visually, and then we had this new vector. And we really had no way of expressing it without drawing it. But when we write things as multiples of the unit vectors. We don't have to draw it. And it's actually very easy to add vectors. And how do we do it? We just add the x components, and we add the y components. So we said that these two vectors, a plus b, these little weird arrows on top, that's just saying that those are vectors. That's equals. So it's minus 3, plus 2i, and I'm going to arbitrarily switch colors, because it's getting monotonous. Plus 2 plus 4j. We just added the x components, or the multiples of i. And we added the y components, or just the multiples of j. Because i was the unit vector in the x direction, and j was the unit vector in the y direction. And we get, what's minus 3 plus 2? That's minus 1. We get minus 1i. That could just be minus i. But I'll write the 1 because we're just getting warmed up with unit vectors. So minus 1i plus 6j. And when I did that, you might say, well, Sal, I can't just take your word for it. Because you seem not someone who should be believed blindly. So I think that's a valid opinion to have. So I will show you that this works, by adding the vectors visually. So let's draw it. And I think this will give you a little better sense of unit vectors generally. Let me draw the axes. So that's my y-axis. Let me draw my x-axis. I have to make sure have enough space to draw the unit vectors that we've drawn, or to draw the vectors that we've drawn. Just to show that the axes go on forever, I have to draw that arrow. All right, so let's say this is 1, 2, 3. This is 1, 2, 3, 4. And I draw 1, 2, 3, 4, 5, 6. I think we should be able to now add them. I didn't have to waste all this space down here. So let's just first draw the vectors, minus 3i plus 2j. So minus 3i, just this right here, is going to be a vector that looks something like this. So it's just minus 3 times the x vector, so it'll go to the left. Because i is 1 in the positive direction. If we put a negative there, it flips it over. Let me use a different color. So this is minus 3i, and then plus 2j. So plus 2j looks like this. If we were to add those two vectors visually, we can put them head to tails. And the way we can do that, we can either shift this vector up like this, and draw it up here. Or we could shift this vector, and put its tail its vector's head. But either way, let's shift this one up. So if we shifted up like that. Remember, we're just doing the head to tails, visual addition method of vectors. So I just put this tail to this head. And what do we get? So vector a will look like this, and I'm going to do it in the same color as vector a because I have a feeling that this diagram might get complicated. Well, I wanted to use the line tool. OK, so this is vector a. That's what vector a looks like. And so we worked backwards. I gave you the x component and the y component. And then I added them together by doing the head to tails method, and so this is what vector a would look like. And, instead of drawing it, a very easy representation is exactly what we did up here, a unit vector notation. And what's vector b look like? So it's 2i-- I'm going to do a completely different color. It's 2i, so it's this vector. 2 times unit vector i. That's this. Plus 4j, 1, 2, 3, 4. So it looks like this. And let's take this one and shift it over to the left, so we can put its tail to the vector's head, so it would look like this. So vector b will look -- I'll do it in red. And I'll use a line tool. Vector b looks like this. I just put its components head to tails, and that's how I got vector b. And if I were to add them visually. I would do it the same way that I added its components. I would put the tail of one vector to the head of the other, and see if you get the resulting vector. So you could do it either way. Let's shift this a vector. Let's shift it in this direction. Remember, vectors, we're just giving the magnitude of direction. We're not necessarily giving a starting point. So you can shift them. You just can't change their orientation or their magnitudes. And that's actually how you add them, you shift them, and put them head to tails. That's when you add them visually. Let's put that a vector up here. So if we have the a vector, it looks something like this. And I want it to work out right. So the a vector looks something like that. And remember, all I did was I took the same vector, and I just shifted it. So that it can start at the head. So its tail can start at the head of the b vector. I just shifted the a vector, so this is still the a vector. By moving the vector around, you haven't changed the vector. I would only change the vector, if I scaled it, if I made it bigger or smaller, if I changed its orientation. And so visually, this is b, this is a, so if I add a to b, the resulting vector, going head to tails-- i'll do it in this green color --would look like this. It would look like that. So here we took all this trouble, and I had to draw these straight lines to visually add these two vectors. This green vector is a plus b. Let's see if this green vector is the same thing that we got here. Let's see if it's the same thing as this. So we got negative 1 times i, so negative 1 is here. And then we have 6j. Let me do it in another color. 6j would look like this. 6j looks like that. You put them heads to tails. And it would be something like this. And that is the green vector. And actually, just so you know, I know it didn't line up perfectly, and that's because I'm not drawing neatly, but these two points should actually be here if I were to have drawn this better. But I know this is very confusing, I had all these colors. But the whole point of it is, I wanted to show that you could visually draws vectors, and then shift them around, and then put them heads to tails. And then get the resulting vector. That's one way to add vectors, there's still no way to analytically represent it. Or you could just write any vector as its x and y components, and then the sum of the vectors is just going to be the sum of the x's and the sum of the y's. And that's a much cleaner, and a much easier, and much less prone to error, way of adding or subtracting two vectors. So hopefully that was convincing. That a plus b really is this vector. If it wasn't, I'm sorry. And I hope I didn't confuse you more. But now that we have this out of the way, and hopefully you're convinced that unit vector notation is useful. We can move on and maybe try to do some of our old projectile motion problems using this notation. And maybe it'll let us to do a little bit of extra stuff with it. See you soon.