# Adding vectors | Vectors and spaces | Linear Algebra | Khan Academy

So I have two

2-dimensional vectors right over here,

vector a and vector b. And what I want to think

about is how can we define or what would be

a reasonable way to define the sum of

vector a plus vector b? Well, one thing that

might jump at your mind is, look, well, each of

these are two dimensional. They both have two components. Why don’t we just add the

corresponding components? So for the sum, why don’t

we make the first component of the sum just a sum of

the first two components of these two vectors. So why don’t we just make

it 6 plus negative 4? Well, 6 plus negative

4 is equal to 2. And why don’t we just

make the second component the sum of the two

second components? So negative 2 plus 4

is also equal to 2. So we start with two

2-dimensional vectors. You add them together,

you get another two 2-dimensional vectors. If you think about it in terms

of real coordinates bases, both of these are

members of R2– I’ll write this down here just

so we get used to the notation. So vector a and vector b

are both members of R2, which is just

another way of saying that these are both two tuples. They are both two-dimensional

vectors right over here. Now, this might make sense just

looking at how we represented it, but how does

this actually make visual or conceptual sense? And to do that, let’s

actually plot these vectors. Let’s try to represent

these vectors in some way. Let’s try to visualize them. So vector a, we could

visualize, this tells us how far this vector

moves in each of these directions–

horizontal direction and vertical direction. So if we put the,

I guess you could say the tail of the vector

at the origin– remember, we don’t have to put

the tail at the origin, but that might make it a little

bit easier for us to draw it. We’ll go 6 in the

horizontal direction. 1, 2, 3, 4, 5, 6. And then negative

2 in the vertical. So negative 2. So vector a could

look like this. Vector a looks like that. And once again,

the important thing is the magnitude

and the direction. The magnitude is represented

by the length of this vector. And the direction

is the direction that it is pointed in. And also just to emphasize,

I could have drawn vector a like that or I could

have put it over here. These are all

equivalent vectors. These are all equal to vector a. All I really care about is the

magnitude and the direction. So with that in mind,

let’s also draw vector b. Vector b in the horizontal

direction goes negative 4– 1, 2, 3, 4, and in the vertical

direction goes 4– 1, 2, 3, 4. So its tail if we

start at the origin, if its tail is at

the origin, its head would be at negative 4, 4. So let me draw that

just like that. So that right over

here is vector b. And once again, vector b

we could draw it like that or we could draw it– let me

copy and let me paste it– so this would also be

another way to draw vector b. Once again, what I

really care about is its magnitude

and its direction. All of these green vectors

have the same magnitude. They all have the same

length and they all have the same direction. So how does the way that

I drew vector a and b gel with what its sum is? So let me draw

its sum like this. Let me draw its sum

in this blue color. So the sum based on this

definition we just used, the vector addition

would be 2, 2. So 2, 2. So it would look

something like this. So how does this make

sense that the sum, that this purple vector

plus this green vector is somehow going to be

equal to this blue vector? I encourage you

to pause the video and think about if

that even makes sense. Well, one way to think about

it is this first purple vector, it shifts us this much. It takes us from this

point to that point. And so if we were to add it,

let’s start at this point and put the green

vector’s tail right there and see where it

ends up putting us. So the green vector, we

already have a version. So once again, we

start the origin. Vector a takes us there. Now, let’s start over

there with the green vector and see where green

vector takes us. And this makes sense. Vector a plus vector b. Put the tail of vector b

at the head of vector a. So if you were to

start at the origin, vector a takes you

there then if you add on what vector b takes you,

it takes you right over there. So relative to the

origin, how much did you– I guess you could say– shift? And once again,

vectors don’t only apply to things

like displacement. It can apply to velocity. It can apply to

actual acceleration. It can apply to a

whole series of things, but when you

visualize it this way, you see that it does

make complete sense. This blue vector,

the sum of the two, is what results where

you start with vector a. At that point right over there,

vector a takes you there, then you take vector b’s

tail, start over there and it takes you to

the tip of the sum. Now, one question you

might be having is well, vector a plus vector

b is this, but what is vector b plus vector a? Does this still work? Well, based on the

definition we had where you add the

corresponding components, you’re still going to

get the same sum vector. So it should come out the same. So this will just be

negative 4 plus 6 is 2. 4 plus negative 2 is 2. But does that make visual sense? So if we start with vector b. So let’s say you

start right over here. Vector b takes you

right over there. And then if you were

to go there and you were to start with vector

a– so let’s do that. So actually, let me make this

a little bit– actually, let me start with a new vector b. So let’s say that that’s our

vector b right over there. And then– actually,

let me give this a place where I’ll have

some space to work with. So let’s say that’s my

vector b right over there. And then let me get a

copy of the vector a. That’s a good one. So copy and let me paste it. So I could put vector a’s

tail at the tip of vector b, and then it’ll take

me right over there. So if I start right over

here, vector b takes me there. And now I’m adding to that

vector a, which starting here will take me there. And so from my original starting

position, I have gone this far. Now, what is this vector? Well, this is exactly

the vector 2, 2. Or another way of

thinking about it, this vector shifts you 2 in

the horizontal direction and 2 in the vertical direction. So either way, you’re going

to get the same result, and that should, hopefully,

make visual or conceptual sense as well.

все векторы одинаковы если у них одинаковое направление и длина.

What programm is he using??

Quando o instrutor fala "membros do R^2", nas legendas em português está escrito "membros do segundo quadrante." Cuidado!

When the instructor says "members of R^2", in the Portuguese subtitles it's written "members of the second quadrant." Beware!

Nice job. What program are you using to do these graphics?

i love your lecture

even though I leave in remote area I always have khan academy…. thank you sir. god bless you and your team

Thank you for the clear explanation ^^

Tanx man you really helped me 🙂

oh lord! ( if u do exist ) plz bless his soul lol

Mind Blowing Tutorial !! (I'm a Law Graduate, but suddenly got reinforced my interest in math by this video)

WORST TUTORIAL EVER, IF YOU CANT EXPLAIN IN IT ONE MINUTE IT SUCKS, NO FORMULA EITHER LOL AND U SAID ALGEBRA IN TITLE STFU

I'm so hungry

Merci 🙂

my intuitions say that he would be pretty good at an fps. lol XD

how do u make these so easy? u r really a great TEACHER.

Thank you so f***ing much, no one answers my curiosity but your videos.

how is vector A the same when it has been plotted differently

Well that helped a lot!!!! 😊😊

How r u adding them when them when they are going in different directions

@4:20 omg mind blown 😂😂

LOVE THE WAY HE ENDED IT.

Thanks a lot, sir, I recently started watching your videos and they do help me a lot with maths and physics,you are really great

I am so crying because my university class teaches vectors without matrices from the first go. Matricies are now part of the subject, but I can't handle this cartesian "remember all the values of all the coordinates" when they look like a=(x,y) or (a1, a2) or (ax, ay) etc (and it only gets worse in 3 dimensions). We only touch on matrices in cross product and using unit vectors. MATRICES ARE SO MUCH MORE CONVENIENT AND EASY TO MANIPULATE ON PAPER, yet they continue to give us annoying component based addition and subtraction like it's easier to actually keep all the values in our head and in such an unorganized manner on paper.

please give me answer"a vector of magnitude 20 is added to a vector of magnitude 25. the magnitude of their sum might be?

But the third side of the triangle can't be equal to the sum of the other two sides…??