Find the vectors T, N, and B at the given point. r(t) = <9 cos t, 9 sin t, 9 ln cos t>, (9, 0, 0)?

1 Answer

• r(t) = <9 cos t, 9 sin t, 9 ln cos t>

  Differentiating,

  r'(t) = <-9 sin t, 9 cos t, 9 * -sin t/cos t>

  .......= <-9 sin t, 9 cos t, -9 tan t>.

  ||r'(t)|| = 9√(sin^2(t) + cos^2(t) + tan^2(t))

  .........= 9√(1 + tan^2(t))

  .........= 9√(sec^2(t))

  .........= 9 sec t.

  So, T(t) = r'(t)/||r'(t)||

  .............= <-9 sin t, 9 cos t, -9 tan t>/(9 sec t)

  .............= <-sin t cos t, cos^2(t), -sin t>.

  Next,

  T'(t) = <-cos^2(t) + sin^2(t), -2 sin t cos t, -cos t>

  ........= <-cos(2t), -sin(2t), -cos t>

  So, ||T'(t)|| = √(1 + cos^2(t)).

  Hence,

  N(t) = T'(t)/||T'(t)||

  .......= <-cos(2t), -sin(2t), -cos t>/√(1 + cos^2(t)).

  -----------

  Letting t = 0:

  T(0) = <0, 1, 0>

  N(0) = <-1, 0, -1>/√2

  B(0) = T(0) x N(0) = <-1, 0, 1>/√2.

  I hope this helps!