Dot product of a vector with the derivative of its unit vector

[hoopsmax25]

Homework Statement

Let c(t) be a path of class C1. Suppose that ||c(t)||>0 for all t.
Show that c(t) dot product with d/dt((c(t))/||c(t)||) =0 for every t.

Homework Equations

I am having trouble with the derivative of (c(t)/||c(t)||) and how to show that when its dotted with c(t) that it always equals zero.

The Attempt at a Solution

I know that c(t) dotted with c'(t) =0 when ||c(t)||= a constant but don't know how to implement this fact for this problem.

 

[Dick]

If d(t)=c(t)/||c(t)||, what is ||d(t)||?

 

[jbunniii]

What is the magnitude of c(t)/||c(t)||?

 

[hoopsmax25]

would it be 1?

 

[Dick]

would it be 1?

Show why it is 1. Then you can drop the '?'.

 

[hoopsmax25]

Ok i get that now. So if d(t)=1, which is a constant, then d(t) dotted with d'(t) =0. Since c(t)/||c(t)|| has the same direction as c(t), we can then plug c(t) in for d(t). Does that make sense to do?

 

[Dick]

Ok i get that now. So if d(t)=1, which is a constant, then d(t) dotted with d'(t) =0. Since c(t)/||c(t)|| has the same direction as c(t), we can then plug c(t) in for d(t). Does that make sense to do?

Makes sense to me.

 

[hoopsmax25]

Awesome. Thanks for the help