2008-03-03

Today's nonsense
- Grief. Is Spiller angry, or is it Dakota that is angry? I don't know what's going on.
- Homework due Wednesday (just like physics).

Unit Tangetn Vectors

Let c be a smooth curve represented by VectorR on an open interval. The unit tangent vector, VectorT(t), at t is defined to be:
VectorT (t) = this is your tangent vector at t = Vectorr ' (t) / || Vectorr (t) || , Vectorr ' (t) != zero-vector. If the tangent is zero, then it'd be zero-over-zero and the position vector was a constant and thus everything is tangent to it so you wouldn't have to find the tangent vector.
The tangent vector is the derivative. It's a unit vector but it's also the derivative.

Find the unit tangent vector to Vectorr(t) = ti + t^{^(2)}j when t=1
On the homework, they will ask you for the unit tangent vector as written with variables, and then evaluated. So they will ask you for VectorT(t) and then they will ask you for VectorT(2) or at whatever t=something they happen to want. So VectorT(2) = (1/sqrt(5)) (i + 2j)

Definition of Principal Unit Normal Vector

Let c be a smooth curve represented by Vectorr(t) on an open interval. If VectorT ' (t) != zero-vector, then the principal unit normal vector at t is defined to be:
VectorN (t) = VectorT ' (t) / || VectorT ' (t) ||
It's like the r became T and the old T became N.

Why is it called "normal vector" if there's no perpendicularity expressed in the equation? Spiller ignored this question. Flat out. Just went on.

Find VectorN(t) and VectorN(1) for VectorR(t) = 3ti + 2t^{^(2)}j

product rule (9+16t^{^(2)})^{^(-1/2)} (3i + 4tj)
---> product rule --> (-1/2)(9+16t^{^(2)})^{^(-3/2)}(32t)(3i+4tj) + (9+16t^{^(2)})^{^(-1/2)}(4j)

There's a shortcut. Obviously this is a perpendicular form. You have -4ti + 3j * 12 / (9+16t^{^(2)})^{^(3/2)} ..... which is just the opposite reciprocal of what you had originally.