Vector Calculus Principle Normal Vector and Binomial Vector
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If T'(t) does not equal 0, it follows that N(t) = T'(t)/||T'(t)||is normal to t(t); is called the principle normal vector. Let a third unit vector that is perpendicular to both T and N be defined by B = T x N; B is called the binomial vector. Together, T, N and B form a right-handed system of mutually orthogonal vectors that may be thought of as moving along a path as shown below.
Show that
a) dB/dt . B = 0
b) dB/dt . T = 0
c) dB/dt = constat * N
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I have used vector calculus to prove the given identities. Solution is in a 2-page word document. I have provided each and every step in this proof, so that the students who decide to down load this answer will understand the process well. They also will be able to use similar techniques in other problems encountered in vector calculus.
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(A) Show dB/dt . B = 0
B . B = 1
Differentiate w.r.t: dB/dt . B + B . dB/dt = 0
2 dB/dt . B = 0
dB/dt . B = 0
(B) Show dB/dt . T = 0
B is normal to T
B . T = 0
dB/dt . T + B . dT/dt = ...
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