1
99:59:59,999 --> 99:59:59,999
So let's say we've got the curve r, defined-

2
99:59:59,999 --> 99:59:59,999
So this is our curve r, it's x of t times i

3
99:59:59,999 --> 99:59:59,999
plus y of t times j it's a curve in 2 dimensions

4
99:59:59,999 --> 99:59:59,999
on the xy-plane, and let's graph it,

5
99:59:59,999 --> 99:59:59,999
just to graph it in a kinda of generalized form.

6
99:59:59,999 --> 99:59:59,999
So that's our y-axis, this is our x-axis, our curve

7
99:59:59,999 --> 99:59:59,999
r might look something like this, it might look something-

8
99:59:59,999 --> 99:59:59,999
Let me draw it a little more of a-

9
99:59:59,999 --> 99:59:59,999
Maybe it looks something

10
99:59:59,999 --> 99:59:59,999
like this maybe that's just part of it,

11
99:59:59,999 --> 99:59:59,999
and as t increases we're going

12
99:59:59,999 --> 99:59:59,999
in that direction right over there.

13
99:59:59,999 --> 99:59:59,999
What I want to do in this video,

14
99:59:59,999 --> 99:59:59,999
and this is really more vector algebra than vector calculus,

15
99:59:59,999 --> 99:59:59,999
is think about, at any given point here,

16
99:59:59,999 --> 99:59:59,999
whether we can figure out a normal

17
99:59:59,999 --> 99:59:59,999
vector, and in particular, a unit normal vector.

18
99:59:59,999 --> 99:59:59,999
Obviously we can figure out

19
99:59:59,999 --> 99:59:59,999
normal vector, you can just divide it's magnitude and you will

20
99:59:59,999 --> 99:59:59,999
get the unit normal vector.

21
99:59:59,999 --> 99:59:59,999
So I want to figure out, at any given point, a vector

22
99:59:59,999 --> 99:59:59,999
that's popping straight out in that direction, and has

23
99:59:59,999 --> 99:59:59,999
a magnitude of 1.

24
99:59:59,999 --> 99:59:59,999
So that would be our unit normal vector.

25
99:59:59,999 --> 99:59:59,999
And to do that

26
99:59:59,999 --> 99:59:59,999
first we will think about what a tangent vector

27
99:59:59,999 --> 99:59:59,999
is and from a tangent vector

28
99:59:59,999 --> 99:59:59,999
we can figure out the normal vector.

29
99:59:59,999 --> 99:59:59,999
It really goes back

30
99:59:59,999 --> 99:59:59,999
to what you might have done in Algebra 1 or

31
99:59:59,999 --> 99:59:59,999
Algebra 2, if you have the slope of a line the

32
99:59:59,999 --> 99:59:59,999
negative reciprocal of that slope is

33
99:59:59,999 --> 99:59:59,999
going to be the slope of the perpendicular line.

34
99:59:59,999 --> 99:59:59,999
We are going to see a very similar thing

35
99:59:59,999 --> 99:59:59,999
when we do it right over here, with the vector-

36
99:59:59,999 --> 99:59:59,999
with this vector algebra.

37
99:59:59,999 --> 99:59:59,999
So the first think I want to think about

38
99:59:59,999 --> 99:59:59,999
how do we construct a tangent line.

39
99:59:59,999 --> 99:59:59,999
Well, you can imagine at some t-some t

40
99:59:59,999 --> 99:59:59,999
this is what our position vector is going to look like

41
99:59:59,999 --> 99:59:59,999
so call that r1-r1 right over there.