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- [Voiceover] So a lot of the ways that we
represent multivariable functions
assume that you're fluent with understanding
how to represent points in three-dimensions
and also how to represent vectors in three-dimensions.
So I thought I'd make a little video here to,
spell out exactly how it is that we describe
points and vectors in three-dimensions.
And before we do that,
I think it will be valuable
if we start off by describing
points and vectors in two-dimensions.
And, I'm assuming if you're learning
about multivariable calculus,
that a lot of you have already learned about this.
And you might be saying
what's the point?
I already know how to represent,
points and vectors in two-dimensions.
But there is a huge value in analogy here,
because as soon as you start to compare
two dimensions and three-dimensions,
you start to see patterns for how it could
extend to other dimensions that you
can't necessarily visualize,
or when it might be useful to think about
one dimension versus another.
So in two-dimensions, if you have some kind of point
just, you know, off sitting there.
We typically represent it,
you've got an x-axis and a y-axis
that a perpendicular to each other.
And we represent this number with a pair.
Sorry, we represent this point with a pair of numbers.
So in this case, I don't know,
it might be something like one, three.
And what that would represent,
is it's saying that you have to move a distance
of one along the x-axis
and then a distance of three up along the y-axis.
So you know this, let's say that's a distance of one,
that's a distance of three,
it might not be exactly that the way I drew it,
but let's say that those are the coordinates.
What this means is that every
point in two dimensional space
can be given a pair of numbers like this
and you think of them as instructions
where it's kind of telling you
how far to walk in one way,
how far to walk in another.
But you can also think of the reverse, right?
Every time that you have a pair of things
you know that you should be thinking two-dimensionaly
and that's actually,
a surprisingly powerful idea that
I don't think I appreciated for a long time
how there's this back and forth between
pairs of numbers and points in space
and it lets you visualize things
you didn't think you could visualize,
or lets you understand things
that are inherently visual just by
kinda going back and forth.
And in three-two-dimensions
there's a similar dichotomy, but between
triplets of points and
points in three-dimensional space.
So, let me just plop down a point
in three-dimensional space here,
and it's hard to get a feel for exactly
where it is until you move things around.
This is one thing that makes three-dimensions hard
is you can't really draw it without
moving it around or showing,
showing a difference in perspective in various ways.
But we describe points like this,
again with a set of coordinates,
but this time it's a triplet.
And this particular point,
I happen to know is one, two, five.
And what those numbers are telling you is
how far to move, parallel to each axis.
So just like with two dimensions,
we have an x-axis,
and a y-axis.
But now there's a third axis that's
perpendicular to both of them,
and moves us into that third-dimension, the z-axis.
And the first number in our coordinate
is gonna tell us how far,
whoop, can't move those guys,
how far we need to move in the x direction
as our first step.
The second number, two in this case,
tells us how far we need to move
parallel to the y-axis, for our second step.
And then the third number tells us how far up
we have to go to get to that point.
And you can do this for any point
in three-dimensional space, right?
Any point that you have you can
give the instructions for how to move
along the x, and then how to move parallel to the y
and how to move parallel to the z
to get to that point,
which means there's this back and forth
between triplets of numbers and points in 3-D.
So whenever you come across a triplet of things,
and you'll see this in the next video
when we start talking about three-dimensional graphs,
you'll know, just by virtue of the
fact that it's a triplet
"Ah, yes, I should be thinking in three-dimensions somehow"
just in the same way whenever you have pairs
you should be thinking
ah, this is a very two-dimensional thing.
So, there's another context though
where pairs of numbers come up
and that would be vectors.
So a vector you might represent,
you know you typically it with an arrow.
Oh,
ahh.
Help, help! (chuckling)
So vectors,
So vectors we typically represent some kind of arrow,
let's, this arrow nice color.
An arrow.
And if it's a vector from the origin to a simple point,
the coordinates of that vector
are just the same as those of the point.
And the convention is to write
those coordinates in a column.
You know, it's not set in stone,
but typically if you see numbers in a column
you should be thinking about it as a vector,
some kind of arrow.
And if it's a pair with parenthesis around it
you just think about it as a point.
And even though, you know, both of these
are ways of representing the same pair of numbers,
the main difference is that a vector
you could have started at any point in space
it didn't have to start in the origin.
So if we have that same guy,
but you know if he starts here
and he still has a rightward component of one
and an upward component of three,
we think of that as the same vector.
And typically these are representing motion of some kind
whereas points are just representing
like actual points in space.
And the other big thing that you can do
is you can add vectors together.
So, you know, if you had another,
let's say you have another vector
that has a large x component
but a small negative y component, like this guy.
And what that means is that you can kind of add
like, imagining that second vector started
at the tip of the first one
and then however you get from the origin
to the new tip there,
that's gonna be the resulting vector, so
I'd say this is, this is the sum of those two vectors.
And you can't really do that with points as much.
In order to think about adding points
you end up thinking about them as vectors.
And the same goes with three-dimensions.
For a given point, if you draw an arrow
from the origin up to that point,
this arrow would be represented
with that same triplet of numbers,
but you typically do it in a column,
I call this a column vector.
That's not three, that's five.
And the difference between the point
and the arrow is you can think of,
you know the arrow or the vector
is starting anywhere in space,
it doesn't really matter,
as long as it's got those same components
for how far does it move parallel to the x,
for how far does it move parallel to the y-axis,
and for how far does it move parallel to the z-axis.
So in the next video,
I'll show how we use these three-dimensions
to start graphing multivariable functions.