r(t)= 1)find T(t) at (1,0,0) and explain what this tells you about the motion of particle on the curve at this point 2)find N(t) 3) find an equation of the line tangent to r(t) at this point. 4) find an equation of a plane normal to r(t) at this point. 5) find the curvature at point (1,0,0)

Question

r(t)=<cost,sint,lncost>
1)find T(t) at (1,0,0) and explain what this tells you about the motion of particle on the curve at this point
2)find N(t)
3) find an equation of the line tangent to r(t) at this point.
4) find an equation of a plane normal to r(t) at this point. 
5) find the curvature at point (1,0,0)

anonymous · Answer

@TuringTest n @Avva   can u guys help me out with this question ?

anonymous · Answer

sorry Idk how to solve this :(

turingtest · Answer

this is just a series of formulas...

anonymous · Answer

@Avva no problem :)

turingtest · Answer

here is the unit tangent vector formula$$\vec T(t)=\frac{\vec r'(t)}{\|\vec r'(t)\|}$$you should be able to guess what the tangent of a position curve represents

turingtest · Answer

the unit normal vector has the formula$$\vec N(t)=\frac{\vec T'(t)}{\|\vec T'(t)\|}$$

anonymous · Answer

for T(t) at (1,0,0) i have $$(0,1/\sqrt{2},0)$$ fot T(t) 
how do i know the motion of a particle ?

turingtest · Answer

the derivative of position is velocity, so that is the direction of the velocity vector of the particle when it is at (1,0,0)

anonymous · Answer

T(t) = <-sint, cost,-tant>/$$\sqrt{1+\sec ^{2}}$$

turingtest · Answer

I have to go, but you are not the only one working on this problem right now why don't you move over to this post and work with amistre64, he can help you :) http://openstudy.com/study#/updates/50ad2c69e4b09749ccabad2f

anonymous · Answer

@TuringTest : hey thank you so much !!

amistre64 · Answer

using what we did for snafus post ....

amistre64 · Answer

tangent line = point + n<tangent vector>

amistre64 · Answer

and the plane is just the point stuck into the a plane equation using the normal vector

anonymous · Answer

wait i am still solving for T(t) and i get <-sin(t)/sec(t),cost/sect,-tant/sect>
is it right so far ?

amistre64 · Answer

1/sec = cos

amistre64 · Answer

so yes so far :)

anonymous · Answer

ok thanks 
m still working on the furture steps !!

anonymous · Answer

now it asks to explain abt the motion of particle 
how would i know that ?

amistre64 · Answer

a vector define a direction and a magnitude

anonymous · Answer

so (0,1,0)is the motion of particle ?

amistre64 · Answer

in what direction is the particle moving?  parallel to the Tangent vector, yes

amistre64 · Answer

the Normal vector tells us in which direction the particle is turning as it moves|dw:1353531203348:dw|