Are These Vector Function Statements True or False?

[zhuyilun]

Homework Statement

True or False:
a. if k(t)=o, the curve is a straight line
b. if the magnitude of r(t)=1 for all t then r'(t) is orthogonalo to r(t)
c. different parametrizations of the same curve result in identical tangent vectors at a given point

Homework Equations

The Attempt at a Solution

a. k(t) is the curvature, which means T'(t) is zero, but what does T'(t)= 0 tells me?
b. if the magnitude of r(t)=1, is the r(t) just a circle/sphere? if r(t) is a sphere, is r'(t) orthogonal to r(t)?
c. i think its true, but i don't know why

 

[LCKurtz]

"k(t) is the curvature, which means T'(t) is zero, but what does T'(t)= 0 tells me?"

It tells you that T(t) is a constant vector. What does that tell you?

If r(t) is a curve, how could it be a sphere?
Does a curve on the surface of a sphere have to be a circle?

 

[zhuyilun]

what about the last question? i think it is right, but i don't know why

 

[Dick]

For b, the magnitude of r(t) is sqrt(r(t).r(t))=1. DIfferentiate that expression. Squaring both sides first makes it a little easier. c is false. Let r(t)=(t,0) and r(t)=(-t,0). What's the tangent vector in each case?

 

[zhuyilun]

For b, the magnitude of r(t) is sqrt(r(t).r(t))=1. DIfferentiate that expression. Squaring both sides first makes it a little easier. c is false. Let r(t)=(t,0) and r(t)=(-t,0). What's the tangent vector in each case?

r(t)=(t,0) and r(t)=(-t,0) won't give you the same cure. i think (-t,0) is not the reparametrization for (t,0)

 

[Dick]

r(t)=(t,0) and r(t)=(-t,0) won't give you the same cure. i think (-t,0) is not the reparametrization for (t,0)

r(t)=(t,0) and r(t)=(-t,0) for t in R are both parametrizations of the curve y=0 in the x-y plane. So for that matter is (t^3,0). Unless you have a much more specific notion of 'reparametrization' in mind. And by 'tangent' do you mean the unit tangent or just r'(t)?