We’ve already seen

that you can visually represent a vector as an arrow,

where the length of the arrow is the magnitude of the vector

and the direction of the arrow is the direction of the vector. And if we want to represent

this mathematically, we could just think

about, well, starting from the tail of

the vector, how far away is the head of the vector

in the horizontal direction? And how far away is it in

the vertical direction? So for example, in the

horizontal direction, you would have to

go this distance. And then in the

vertical direction, you would have to

go this distance. Let me do that in

a different color. You would have to go this

distance right over here. And so let’s just say

that this distance is 2 and that this distance is 3. We could represent

this vector– and let’s call this vector v. We

could represent vector v as an ordered list or a

2-tuple of– so we could say we move 2 in the

horizontal direction and 3 in the vertical direction. So you could represent

it like that. You could represent

vector v like this, where it is 2

comma 3, like that. And what I now want

to introduce you to– and we could come up with

other ways of representing this 2-tuple– is

another notation. And this really

comes out of the idea of what it means to

add and scale vectors. And to do that,

we’re going to define what we call unit vectors. And if we’re in

two dimensions, we define a unit vector for

each of the dimensions we’re operating in. If we’re in three dimensions,

we would define a unit vector for each of the three dimensions

that we’re operating in. And so let’s do that. So let’s define a unit vector i. And the way that we denote

that is the unit vector is, instead of putting

an arrow on top, we put this hat on top of it. So the unit vector

i, if we wanted to write it in this

notation right over here, we would say it only goes 1 unit

in the horizontal direction, and it doesn’t go at all

in the vertical direction. So it would look

something like this. That is the unit vector i. And then we can define

another unit vector. And let’s call

that unit vector– or it’s typically

called j, which would go only in the

vertical direction and not in the horizontal direction. And not in the

horizontal direction, and it goes 1 unit in

the vertical direction. So this went 1 unit

in the horizontal. And now j is going to go

1 unit in the vertical. So j– just like that. Now any vector, any

two dimensional vector, we can now represent as a sum of

scaled up versions of i and j. And you say, well,

how do we do that? Well, you could imagine

vector v right here is the sum of a vector

that moves purely in the horizontal direction

that has a length 2, and a vector that moves purely

in the vertical direction that has length 3. So we could say

that vector v– let me do it in that

same blue color– is equal to– so if we want

a vector that has length 2 and it moves purely in

the horizontal direction, well, we could just scale

up the unit vector i. We could just

multiply 2 times i. So let’s do that– is equal

to 2 times our unit vector i. So 2i is going to

be this whole thing right over here or

this whole vector. Let me do it in

this yellow color. This vector right over

here, you could view as 2i. And then to that, we’re going to

add 3 times j– so plus 3 times j. Let me write it like this. Let me get that color. Once again, 3 times

j is going to be this vector right over here. And if you add this yellow

vector right over here to the magenta vector,

you’re going to get– notice, we’re putting the tail

of the magenta vector at the head of

the yellow vector. And if you start at the

tail of the yellow vector and you go all the way to the

head of the magenta vector, you have now constructed

vector v. So vector v, you could represent it as a

column vector like this, 2 3. You could represent

it as 2 comma 3, or you could represent it as

2 times i with this little hat over it, plus 3 times j,

with this little hat over it. i is the unit vector in

the horizontal direction, in the positive

horizontal direction. If you want to go

the other way, you would multiply it by a negative. And j is the unit vector

in the vertical direction. As we’ll see in future

videos, once you go to three dimensions,

you’ll introduce a k. But it’s very natural to

translate between these two things. Notice, 2, 3– 2, 3. And so with that, let’s actually

do some vector operations using this notation. So let’s say that I

define another vector. Let’s say it is vector b. I’ll just come up with

some numbers here. Vector b is equal to negative 1

times i– times the unit vector i– plus 4 times the unit vector

in the horizontal direction. So given these two

vector definitions, what would the would be the

vector v plus b be equal to? And I encourage you to pause

the video and think about it. Well once again,

we just literally have to add

corresponding components. We could say, OK,

well let’s think about what we’re doing in

the horizontal direction. We’re going 2 in the

horizontal direction here, and now we’re going negative 1. So our horizontal

component is going to be 2 plus negative

1– 2 plus negative 1 in the horizontal direction. And we’re going to multiply

that times the unit vector i. And this, once

again, just goes back to adding the corresponding

components of the vector. And then we’re going to have

plus 4, or plus 3 plus 4– And let me write it that way–

times the unit vector j in the vertical direction. And so that’s going

to give us– I’ll do this all in this one color–

2 plus negative 1 is 1i. And we could literally

write that just as i. Actually, let’s do that. Let’s just write that as i. But we got that from 2

plus negative 1 is 1. 1 times the vector is just

going to be that vector, plus 3 plus 4 is 7– 7j. And you see, this is exactly

how we saw vector addition in the past, is that we

could also represent vector b like this. We could represent it

like this– negative 1, 4. And so if you were

to add v to b, you add the corresponding terms. So if we were to add

corresponding terms, looking at them as column

vectors, that is going to be equal to 2 plus

negative 1, which is 1. 3 plus 4 is 7. So this is the exact same

representation as this. This is using unit

vector notation, and this is representing

it as a column vector.

Thank you from a struggling Scot with my exam tomorrow lol! Super helpful

Don't we add v and b by pythagorus theorem?

what is the point of this other than more confusing notation

Your explanation is so much better than our teacher

Thanks a lot you don't know I was in a big trouble finding this simple thing thanks again

This video but every time he changes the color, it gets faster.

Is vector in (i j k) form always a position vector or it can be between any two coordinate points ?

Is vector v here is a unit vector

Very well explained sir..

I have a question. For labeling vector as a coordinate, you put v=(2, 3). But when you wrote vectors with unit vector i and j, why did you do 2i+3j, not (2i, 3j)?

Superb

why do you use a cap for i and j but a arrow for v and b??

What if the x and y components are negative?