Mindscrape
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Okay, so I was asked to find all the things listed in the topic title given the equation:
[tex]r(t)=(cos^{3}t)\vec{i} + (sin^{3}t)\vec{j}[/tex]
Now this is a lot of work, especially when it comes to finding the torsion [tex]\tau = - \frac{d \vec{B}}{ds} \cdot \vec{N}[/tex] a total of four derivitives. Maybe I am missing something about how these all relate, and I could somehow get around all the work. I would put up my work so far, but that would take a really long time.
Basically, what I have done is find the tangent vector, so v/mag(v); the normal vector, derivative(tangent)/mag(tangent); the binormal, TxN; and now I got to Torsion (d(binormal)/dt)/mag(v) dot N. Torsion looks like it will be a long equation with my method.
Thoughts?
[tex]r(t)=(cos^{3}t)\vec{i} + (sin^{3}t)\vec{j}[/tex]
Now this is a lot of work, especially when it comes to finding the torsion [tex]\tau = - \frac{d \vec{B}}{ds} \cdot \vec{N}[/tex] a total of four derivitives. Maybe I am missing something about how these all relate, and I could somehow get around all the work. I would put up my work so far, but that would take a really long time.
Basically, what I have done is find the tangent vector, so v/mag(v); the normal vector, derivative(tangent)/mag(tangent); the binormal, TxN; and now I got to Torsion (d(binormal)/dt)/mag(v) dot N. Torsion looks like it will be a long equation with my method.
Thoughts?
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