Unit tangent vectors

**Unenlightened** · December 1st 2009, 08:40 AM

Let $r(t) = 3 sin({\it t}{\bf i}) + 3 cos ({\it t}{\bf j}) + 4{\it t}{\bf k}$ . At $t = 0$ , find
(i) the unit tangent vector T,
(ii) the unit normal vector N,
(iii) the curvature κ,
(iv) the binormal vector B,
(v) the torsion $\tau$

I'll come back to this one with my thoughts on it.
For the moment, I must go catch some shuteye...

**adkinsjr** · December 1st 2009, 09:44 AM

Originally Posted by Unenlightened

Let $r(t) = 3 sin({\it t}{\bf i}) + 3 cos ({\it t}{\bf j}) + 4{\it t}{\bf k}$ . At $t = 0$ , find
(i) the unit tangent vector T,
(ii) the unit normal vector N,
(iii) the curvature κ,
(iv) the binormal vector B,
(v) the torsion $\tau$

I'll come back to this one with my thoughts on it.
For the moment, I must go catch some shuteye...

If you were given this problem, surely you were shown how to find these. There's really nothing to it. It's just direct application of the theorems and definitions. You should know how to find the derivative of a vector $r'(t)$ and the unit tangent vector at any point $t$ is just $T(t)=\frac{r'(t)}{\mid r'(t) \mid}$ .

The curvature $k=\mid\frac{dT}{ds}\mid$

Just find the definitions in the book and apply them. Like I said, you don't really have to think about much to do these excercises.

**Unenlightened** · December 2nd 2009, 04:39 AM

Originally Posted by adkinsjr

If you were given this problem, surely you were shown how to find these.

Aye, I know. I wasn't given the problem though - I (perhaps foolishly) offered to help some friends who were struggling with their maths. I'll go wikipedia the definitions presently.

Cheers for the reply anyway.

**HallsofIvy** · December 2nd 2009, 06:28 AM

Originally Posted by Unenlightened

Let $r(t) = 3 sin({\it t}{\bf i}) + 3 cos ({\it t}{\bf j}) + 4{\it t}{\bf k}$ . At $t = 0$ , find
(i) the unit tangent vector T,

Differentiate with respect to t. The normal vector is the vector of unit length pointing in the direction of that derivative vector.

(ii) the unit normal vector N,

The length of the derivative in (i), is the arclength as a function of t. If you can solve for t as a function of s, differentiate the unit tangent vector with respect to s to get a normal vector. Divide by the length of that to get the unit normal vector.

(iii) the curvature κ,

The length of the derivative of the unit tangent vector, with respect to s, is the curvature.

(iv) the binormal vector B,

The binormal vector is the cross product of the unit tangent and unit normal vectors. Divide by its length to get the unit binormal vector if that was asked.

(v) the torsion $\tau$

And the length of that binormal vector is the torsion.

I'll come back to this one with my thoughts on it.
For the moment, I must go catch some shuteye...

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