Find T,N,B Vectors & Curvature of Curve x=-4ty=-t2z=-2t3 at t=1

[FreedayFF]

Homework Statement

Find the unit tangent, normal and binormal vectors T,N,B , and the curvature of the curve
x=−4t y=−t² z=−2t³ at t=1.

Homework Equations

The Attempt at a Solution

I found T=(-4/sqrt(56),-2/sqrt(56),-6/sqrt(56)) which is correct. But I keep getting N wrong? PLease help me, thank you!

 

[LCKurtz]

Looks good as far as it goes. Hard to say what you are doing wrong unless you show us what you are doing to get N.

 

[FreedayFF]

N = T'/|T'|
so the first term should be 0; but the webwork keeps denying that answer.

 

[FreedayFF]

N = (0,-2/sqrt(148),-12/sqrt(148)) and that gives a wrong answer

 

[cronxeh]

Not 100% sure of this (im at work), but you get the train of thought:
Try the answers in bold
r(t) = (-4t)i + (-t^2)j + (-2*t^3)k

v(t) = r'(t) = (-4)i -(2*t)j - (6*t^2)k
||v(t)|| = sqrt(2*(4+t^2+9*t^4))

T(t) = v(t)/||v(t)|| =((-4)i + (-2*t)j + (-6t^2))/sqrt(2*(4+t^2+9*t^4))

Evaluate at t=1

T(1) = ((-4)i + (-2)j + (-6)k)/(6503/869) = 869*(-4i-2j-6k)/6503

N(t) = T'(t)/||T'(t)|| = diff(T,t)/length(diff(T,t))

N(t) = 0i - (2/3) j - (4*t)k
evaluate at t=1

N(1) = 0i - 2/3j -4k

B=T x N = [ 4*t^2, -16*t, 8/3]
at t=1

B(1) = 4i - 16j + 8/3k

 

[FreedayFF]

I believe your answers are incorrect, as I have all T,B, curvature correct in the attached image above. Your T, however, has the same i value as mine, which is 0 and incorrect.

 

[cronxeh]

T=[-0.53454,-0.26727,-0.80181]
N=[0.81053,0.10665,-0.57592]
B=[0.23942,-0.95771,0.15962]

Maple rules.

 

[FreedayFF]

Okay Maple rules do give correct answers. Thanks. But I don't understand why they would give different answers to the "conventional" methods. I'm confused about when to use Maple rules and when not?

 

[cronxeh]

Okay Maple rules do give correct answers. Thanks. But I don't understand why they would give different answers to the "conventional" methods. I'm confused about when to use Maple rules and when not?

I made a mistake earlier while using the conventional method. The correct way is:

N(t) = T'(t)/||T'(t)|| = (r'(t) x (r''(t) x r'(t)))/(||r'(t)||*||r''(t) x r'(t)||) = <304/(8*sqrt(157)*sqrt(14)), 40/(8*sqrt(157)*sqrt(14)), -216/(8*sqrt(157)*sqrt(14))>

DUH. It all checks out.