Motion defined by a parametric eqn

[Locoism]

Homework Statement

For the curve:
r(t) = ⟨$\frac{1}{2}$t⁵, $\frac{1}{3}$t⁵, $\frac{1}{6}$t⁵⟩,

Find the arc length s(t)
Find the unit tangent T(t) and T(1)
Find the principle unit normal N(t) and N(1)
Find the binormal vector B(t) and B(1)

Homework Equations

T(t) = $\frac{r'(t)}{|r'(t)|}$

N(t) = $\frac{T'(t)}{|T'(t)|}$

The Attempt at a Solution

I found s(t) as $\frac{1}{6}$t⁵$\sqrt{10}$

and r'(t) = <$\frac{5}{2}$t⁴, $\frac{5}{3}$t⁴, $\frac{5}{6}$t⁴>

But now if I calculate T(t) I get $\frac{1}{\sqrt{10}}$<3, 2, 1>
I'm sure this is wrong because first of all the question wouldn't ask for T(1), and secondly because now T'(t) is a zero vector, which makes N(t) a zero vector, and B(t) likewise.
Have I made a mistake or is the question just asking for some really trivial stuff?

 

[Dick]

Well, the curve r(t) is a straight line, right? So it would make sense T(t) is a constant. And N(t) is going to be undefined, not zero. So that does make it a pretty strange question. Maybe there's a typo in r(t)??

 

[Locoism]

Hm ok then, maybe the question has a mistake... But why would N be undefined and not zero?
Also the equation of the osculating plane at t=1 would also be undefined?
Thank you

 

[Dick]

Hm ok then, maybe the question has a mistake... But why would N be undefined and not zero?
Also the equation of the osculating plane at t=1 would also be undefined?
Thank you

If T'(t)=0 then N(t)=T'(t)/|T'(t)| is 0/0. That's undefined. There's no unique normal. ANY vector perpendicular to the line is a normal. Nope, no osculating plane either.

 

[Locoism]

Ah ok thanks a lot!