Learn about Unit Vector Notation - part 2

Khan Academy Presents: 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

Welcome back. In the last video, I just you know, at the end of the video like I always do and the thing to confuse you. I told you that if I had two vectors and let me just make up some new once just so I can draw them visually in a second or two. Let’s call the first vector A with a different color. This is toothpaste to color is getting monotonous. Let me do something interesting. So let’s call the first vector A and I don’t know. Let’s say it’s – let me put some and make it interesting. Let me say it’s minus three times the unit vectors i plus two times the unit vector j and then I have vector b and that is equal to I don’t know. That’s its 2i so two times the unit vector i plus I don’t know. Let me say four times the unit vector j. In the last video I said — well the whole reason why this unit vector notation is even – one of the reasons what we’ll see that there are many reasons why it’s usable. 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 have 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 is 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 + B and these little weird arrows on top that’s just saying that those are vectors to that equals – this should have been equal sign – not now equals that’s equals. So it’s minus three plus two i and I’m going to arbitrary switch colors because it’s getting monotonous— plus two plus four j. we just added the X components or the multiples of i and we added the Y components or just some multiples of j, right? Because i was the unit vector in the x direction and j was the unit vector and the y direction and we get, what’s -3+2? That’s -1 so we get -1i or that could just be -i but I’ll write the one just because we’re just getting warmed up with the unit vectors so -1i+6j. When I did that, you might say we’ll Sal you know, I can’t just take their word for it because you seen kind of you know, – not someone who should be believed blindly. And I think that’s a valid opinion to have so I will show you that this works by doing, by adding the vectors visually so let’s draw it and I think this will give you a better sense of a unit vectors generally. Let me draw the axis so that’s my x, that’s my y axis and we draw like x axis—to make sure have enough space to draw that unit vectors that we’ve drawn and to draw the vectors that we’ve drawn. So that’s just to show that the axis go on forever, let’s draw that arrow. Alright so it’s—. Let’s say this is 1, 2, 3 this is 1, 2, 3, 4 then I’ll draw 1, 2, 3, 4, 5, 6 and I think we should be able to now add them. I didn’t have to waste all the space right here. Let’s just first draw the vectors -3i+2j so -3i, so -3i just this—this right here is going to be a vector that looks something like this. So to go it’s just -3x the X vector so it’ll go to the left because as i is the pos— is one in the positive directions if we put a negative there it flips it over. Let me do it in different color so this is -3i and then + 2j. So +2j looks like this. Alright this is +2j and if we were to add those two vectors visually, we can put them head to tails and the way we could do that, we could either shift to these vector up like this or draw it up here. We could shift this vector and put its tail to this vectors head but either way, so let’s shift this one up so if we shift it 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 just so— but I have feeling that this diagram might get complicated. If I wanted to just use a line tool and undo the line tool. 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.—Instead of drawing it in a very easy representation it’s exactly what we did up here, the 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 the unit vector i that’s this, plus 4j. 4j 1, 2, 3, 4 so it looks like this. Let’s take this one and shift over the last so we can put its tail to this vectors head so it will look like this. So vector B will look, and I’ll do it in red. And I’ll use the line tool vector B looks like this. Alright I’d just put it’s components head to tails and that’s how I get vector B. And if I were to add them visually, I would do the same way that I added to its components. I would put the tail of one vector to the head of the other and seek at the resulting vector. Through we can do it in either way. Let’s shift to this A vector and let’s shift it in this direction and put it. — because remember vectors— we’re just giving a magnitude and direction we’re telling a, we’re not necessarily giving a starting point, so you can shift them. You just can’t change their orientation or the magnitude and that’s actually how you add them, you’ll shift them and put them head to tails. That’s when you add them visually. Let’s put that A vector and let’s put it up here so if we have the A vector that looks something like this, I know my – the A vector looks something like this and I want it to work out right, so the A vector, something like that. Remember all I did is I took the same vector and I just shifted it so that it can start at the head, so it's tail can start at the head of the b vector, so I just shifted the a vector so this is still the a vector. By moving a vector on, you haven’t changed the vector. I would only change the vector if I scaled it, it I made it bigger or smaller or 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 - Let me see how does in this green color. It would look like this. It would look like that right? So here we took all this trouble and I had to draw the straight lines to visually add this two vectors. This green vector is A+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 -1xi, so -1 is like here. And then we have 6j and we do it another color and 6j would look like this. 6j looks like that, you put them heads to tails and it would get you 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 was very confusing, I had all these colors but the whole point of it is, I want you to show that you know – you could visually draw vectors and then have you know, shift them around and then put them heads to tails and then get the resulting vector. That’s when we’d add vectors and there’s still real no way to anal— analytically represented it or you could just write any vector as it's X and Y components and then the sum of the vectors is just going to be the sum of the X’s and then the sum of the Y’s. And that’s a much cleaner and a much easier and a much less prone to error way of adding or subtracting really two vectors. Hopefully, that was convincing that this A+g – A+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 will let us to do a little bit extra stuff with it. See you soon.