Curvature question

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.

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
Homework Helper
Gold Member
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.

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

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

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cronxeh
Gold Member
Not 100% sure of this (im at work), but you get the train of thought:
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

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

Maple rules.

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
Gold Member
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.