Suppose r:[a,b]--r^{3}
unit tangent vector is (dr/dt)/|(dr/dt)=T(t) when we say N(t)=(dT/dt)/|(dT/dt) and say it is orthogonal is not there other vectors with that condition ?

Question

Suppose r:[a,b]-->r^{3}
unit tangent vector is (dr/dt)/|(dr/dt)=T(t) when we say N(t)=(dT/dt)/|(dT/dt) and say it is orthogonal is not there other vectors with that condition ?

anonymous · Answer

Please make your question clearer, what do you mean by ' and say it is orthogonal is not there other vectors with that condition'?

anonymous · Answer

|dw:1365936088776:dw| I mean as T(t) uniqe vector why N is uniqe too ? 
there is other vectors that are normal to T(t)

anonymous · Answer

because N(t) is the normalized one?

anonymous · Answer

I mean there is a plane that is normal or orthogonal to T(t) so other vectors that are on plane can be N(t) but why is it unique ?

anonymous · Answer

Ah I see your point, let me think about it

anonymous · Answer

T(t) is orthogonal to T'(t) for all t right?

anonymous · Answer

Yes

anonymous · Answer

I think it's because the way we choose it. Out of many vectors that are orthogonal to T(t), we pick T'(t), and normalize it to get N(t). That's it.

anonymous · Answer

do we have to check T(t) and N(t) in R^{2} instead of R^{3} ?

anonymous · Answer

check what?

anonymous · Answer

check the problem I mean

anonymous · Answer

About the unicity of such vector right? Nope, you don't have to. (Actually the problem is trivial in R^2 you can just visualize it yourself)

anonymous · Answer

Thanks for help