- #1
Eclair_de_XII
- 1,083
- 91
Homework Statement
##r(t)=(t^2)i+(1+\frac{1}{3}t^3)j+(t-\frac{1}{3}t^3)k##
Find the tangential, normal, and binormal vectors for this TNB frame.
Homework Equations
##T(t)=\frac{v(t)}{|v(t)|}##
##N(t)=\frac{T`(t)}{|T`(t)|}##
##B(t)=T(t)×N(t)##
The Attempt at a Solution
The problem isn't that I don't know how to solve this, it's just that I don't know how to solve this fast enough. I had about forty minutes to figure out this problem, to calculate the normal and tangent components of acceleration, and to find the velocity and acceleration of an unrelated polar equation. In the end, I didn't have enough time to finish all of the quiz. Here's what I did...
##v(t)=(2t)i+(t^2)j+(1-t^2)k##
##|v(t)|=\sqrt{4t^2+t^4+1-2t^2+t^4}=\sqrt{2t^4+2t^2+1}##
##T(t)=\frac{2t}{\sqrt{2t^4+2t^2+1}}i+\frac{t^2}{\sqrt{2t^4+2t^2+1}}j+\frac{(1-t^2)}{\sqrt{2t^4+2t^2+1}}k##
And here's the time-consuming part that took up much of the allotted time.
##T_x`(t)=\frac{2(\sqrt{2t^4+2t^2+1})-2t(\frac{1}{2})(8t^3+4t)(2t^4+2t^2+1)^{-\frac{1}{2}}}{2t^4+2t^2+1}##
##T_x`(t)=\frac{4t^4+4t^2+2-8t^4-4t^2}{(2t^4+2t^2+1)^{\frac{3}{2}}}=\frac{2-4t^4}{(2t^4+2t^2+1)^{\frac{3}{2}}}##
Then I had to do this for the y- and z-components, then the magnitude, and it just drained away so much time. I need a faster way to compute N(t), and in turn, B(t), in other words.