Given the vector r(t) = { cosT, sinT, ln (CosT) } and point (1, 0, 0) find vectors T, N and B at that point.Vector T is the unit tangent vector, so the derivative r(t) is needed.

banganX

banganX

Answered question

2021-01-28

Given the vector r(t)=cosT,sinT,ln(cosT) and point (1, 0, 0) find vectors T, N and B at that point.

Vector T is the unit tangent vector, so the derivative r(t) is needed.

Vector N is the normal unit vector, and the equation for it uses the derivative of T(t).

The B vector is the binormal vector, which is a crossproduct of T and N.

Answer & Explanation

Ezra Herbert

Ezra Herbert

Skilled2021-01-29Added 99 answers

The derivative of the vector is,
r(t)={sin5,cost,tant}
The unit tangent vector, T at the given point is obtained as follows.
T(t)=r(t)||r(t)||
T(t)=sint,cost,tant(sint)2+(cost)2+(tant2
T(t)=sintsect,costsect,tantsect
T(t)|1,0,0=0.45,1,0
The derivative of the unit tangent vector, T is,
T(t)=costtantsintsect,sin2t,cost
The normal unit vector, N at the given point is obtained as follows.
N(t)=T(t)||T(t)||
N(t)=costtantsintsect,sin2t,cost(cost+tantsint)2+sin2(2t)sec2(t)+1sec(t)
N(t)|1,0,0=0.366,0,0.70
The binormal vector, B at the given point is obtained as follows.
B(t)=T(t)xxN(t)
B(t)|1,0,0=[ijk0.45100.36600.70]
=(0.70)i(0.315)j(0.366)k

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