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

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

November 19, 2019 25 By Kody Olson


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.