00:01
Okay, in steps in problem three, we're given this is the derivative of our position vector with respect to time.
00:10
We're given this initial position.
00:13
You want to find r of t.
00:16
So we integrate, and we can integrate component by component.
00:22
So we do each of the three components separately.
00:26
And then we get a, right now, unknown constant vector.
00:32
That's our constant of integration.
00:35
Because this is a vector formula, we have really three components of the constant.
00:45
And then we can say, let's see what r of 0 is.
00:50
So we plug in t equals zero here.
00:54
E to the 0 is 1, so that's a 4.
00:57
And cosine of 0 is also 1, so that's a minus 1.
01:04
And that has to be plus our constant vector c has to equal r.
01:09
Initial position which was 1, 2, 3.
01:15
And then we can solve that for c, which is 1 minus 2 ,4.
01:29
That's it.
01:30
So problem four, we're given r of t.
01:40
So you might wanna recognize that this, this parametric equation is a parametric equation for a circle of radius b that's centered at the origin.
01:55
First of all, we want to figure out what that equation for r double dot is.
02:01
So the second derivative of the cosine of omega -t is minus omega -square times the cosine, and the second derivative of sign with respect to t is minus omega -squared times the sine.
02:18
Okay.
02:19
And so we can notice that it'll satisfy this.
02:29
Equation with a is equal to omega squared.
02:37
Now for part b, we want to do some of the curve theory kind of analysis.
02:45
So the tangent vector we get by taking the time derivative of r.
02:52
It looks like this.
02:54
But what we really need for curve theory is the unit tangent.
03:03
Vector.
03:07
So if i calculate the magnitude of t, it looks like this.
03:14
And what's underneath the square root there is sine squared plus cosine squared, which is one...