Position vector functions and derivatives
Position Vector Valued Functions Using a position vector valued function to describe a curve or path
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- All of the work we've been doing so far with line
- integrals has been with scalar functions or
- scalar-valued functions.
- And when I say that, that just means you give me an x and a y
- and you evaluate the function at that x and y, and you
- get another scalar value.
- You get just a number.
- You don't get a vector.
- So what I want to do in this video is start to get ourselves
- warmed up with regards to vectors so that we can
- understand what it means to take a line integral with
- vector-valued functions.
- So let me write down some vector-valued functions.
- Actually, even a better place to start, let me draw a curve
- or let me describe a curve.
- So let's put that little f of x, y to the side.
- We can ignore it for now.
- Let's say I have some curve c and it's described, it can be
- parameterized-- I can't say that word-- as let's say, x is
- equal to x of t, y is equal to some function y of t.
- And let's say that this is valid for t is between a and b.
- So t is greater than or equal to a and then,
- less than or equal to b.
- So if I were to just draw this on-- let me see-- I
- could draw it like this.
- I'm staying very abstract right now.
- This is not a very specific example.
- This is the x-axis.
- This is the y-axis.
- My curve-- let's say this is when t is equal to a.
- And then the curve might do something like this.
- I don't know what it does.
- Let's say it's over there.
- This is t is equal to b.
- This actual point right here will be x of b.
- That would be the x-coordinate.
- You evaluate this function at b and y of b.
- And this is, of course, when t is equal to a.
- The actual coordinate in r2 on the Cartesian coordinates will
- be x of a, which is this right here.
- And then, y of a, which is that right there.
- And we've seen that before.
- That's just a standard way of describing a parametric
- equation or curve using 2 parametric equations.
- What I want to do now is describe this same exact curve
- using a vector-valued function.
- So if I define a vector-valued function-- and if you don't
- remember what those are, we'll have a little bit
- of review here.
- Let me say I have a vector-valued function, r,
- and I'll put a little vector arrow on top of it.
- And a lot of textbooks, they'll just bold it and they'll
- leave scalar-valued functions unbolded.
- But it's hard to draw bold, so I'll put a little
- vector on top.
- And let's say that r is a function of t.
- And these are going to be position vectors.
- And I'm specifying that because, in general, when
- someone talks about a vector, this vector and this vector
- are considered equivalent.
- As long as they have the same magnitude and direction, no one
- really cares about what their start and end points are as
- long as their direction's the same and their
- length is the same.
- But when you talk about position vectors you're saying
- no, these vectors are all going to start at 0, at the origin.
- And when you say it's a position vector, you're
- implicitly saying this is specifying a unique position.
- In this case, it's going to be in two-dimensional
- space, but it could be in three-dimensional space.
- Or really, even four, five, whatever-- n dimensional space.
- So when you say it's a position vector, you're literally
- saying, OK, this vector literally specifies
- that point in space.
- So let's see if we can describe this curve as a position
- vector-valued function.
- So we could say r of t.
- Let me switch back to that pink color.
- This can stay in green.
- Is equal to x of t times the unit vector in the x direction.
- The unit vector gets a little caret on top-- a little hat.
- That's like the arrow for it.
- That just says it's a unit vector.
- Plus y of t times j.
- If I was dealing with a curve in three dimensions I would
- have plus z of t times k.
- But we're dealing with two dimensions right here.
- And so the way this works is you're just taking your-- well,
- for any t and still, we're going to have t is greater
- than or equal to a and then, less than or equal to b.
- And this is the exact same thing as that.
- Let me just redraw it.
- So let me draw our coordinates.
- Our coordinates right here, our axes.
- So that's the y-axis and this is the x-axis.
- So when you evaluate r of a, that's our starting point.
- So let me do that.
- So r of a-- maybe I'll do it right over here.
- Our position vector-valued function evaluated at t is
- equal to a, is going to be equal to x of a times our unit
- vector in the x direction.
- Plus y of a times our unit vector in the vertical
- direction, or in the y direction.
- And what's that going to look like?
- Well, x of a is this thing right here, so it's x of
- a times a unit vector.
- You know, maybe the unit vector is this long.
- It has length 1, so now we're just going to have a length
- of x of a in that direction.
- And then, same thing in y of a.
- It's going to be y of a length in that direction.
- But the bottom line, this vector right here-- if you add
- these scaled values of these two unit vectors, you're going
- to get r of a looking something like this.
- it's going to be a vector that looks something like that.
- Just like that.
- It's a position vector.
- That's why we're nailing it at the origin, but drawing
- it in standard position.
- And that right there is r of a.
- Now what happens if a increases a little bit?
- What is r of a plus a little bit?
- And I don't know, we could call that r of a plus
- delta or r of a plus h.
- I'll do it in a different color.
- Let's say we increase a a little bit. r of a
- plus some small h.
- Well, that's just going to be x of a plus h times
- a unit vector i.
- Plus y times a plus h times the unit vector j.
- And what's that going to look like?
- Well, we're going to go a little bit further
- down the curve.
- That's like saying the coordinate x of a plus
- h and y plus a plus h.
- I might be that point right there.
- So it'll be a new unit vector.
- Sorry, it'll be a new vector-- position vector--
- not a unit vector.
- These don't necessarily have length 1.
- That might be right here.
- Let me do that same color as this.
- So it might be just like that.
- So that right here is r of a plus h.
- So you see, as you keep increasing you value of t until
- you get to b, these position vectors-- we're going to keep
- specifying points along this curve.
- So the curve-- let me draw the curve in a different color.
- The curve looks something like this.
- It's meant to look exactly like the curve that I have up here.
- And for example, r of b is going to be a vector
- that looks like this.
- It's going to be a vector that looks like that.
- I want to draw it relatively straight.
- That vector right there is r of b.
- So hopefully you realize that, look, these position vectors
- really are specifying the same points on this curve as this
- original, I guess, straight up parameterization that
- we did for this curve.
- And I just wanted to that as a little bit of review because
- we're now going to break in into the idea of actually
- taking a derivative of this vector-valued function.
- And I'll do that in the next video.
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