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
Unit Vector Notation (part 2)
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- 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
- 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.
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