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### Course: MCAT > Unit 9

Lesson 1: Vectors and scalars# 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.

## Want to join the conversation?

- Why do vectors have to be arranged "head to tales" in order for the addition to work out?(36 votes)
- Think of it like walking,

I walked 3 meters left and 2 forward,

then I walked 2 meters right and 4 forward....

Where am I?

Wouldn't make sense to start both movements at the origin(124 votes)

- 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)
- 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).(4 votes)

- 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 at2:23)(5 votes)
- 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) at5:45what 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)- 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)

- how would you subtract using the head-to-tail method?(4 votes)
- Please watch video for subtraction of vectors via the head-to-tail and the tail-to-tail methods. ~ http://youtu.be/mFu5IPOG5Cw(6 votes)

- can 2 vectors be added if one is in 2 dimension & the other one is in 3 Dimension ?(2 votes)
- Yes, here is an example.
**A**= a**i**+ b**j****B**= d**i**+ e**j**+ f**k****A**+**B**= (a+d)**i**+ (b+e)**j**+ f**k**(4 votes)

- How would u visually subtract the vectors?(2 votes)
- keeping them on the same origin, you can connect their tips.(2 votes)

- how to find an angle between 2 vectors when both have equal magnitudes(2 votes)
- You'll need more information, tha magnitude of two vectors doesn't say the necessary things to discover the angle between they.(3 votes)

- 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)
- You have to take two vectors, put them head to tails unless you use vector notation and the resultant side is the third vector. Put the first, second, and third vector together and you get a complete triangle.(1 vote)

- can you define a unit vector in terms of i and j..(2 votes)
- A unit vector is a vector with magnitude equal to one. It is used to specify a direction, without any value attached to it. In the xy-plane, a unit vector û that makes an angle θ with the x-axis, has the formula: û = i cos θ + j sin θ.(2 votes)

## 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.