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Section 12.8 : Tangent, Normal and Binormal Vectors

For problems 1 & 2 find the unit tangent vector for the given vector function.

  1. \(\vec r\left( t \right) = \left\langle {{t^2} + 1,3 - t,{t^3}} \right\rangle \) Solution
  2. \(\vec r\left( t \right) = t{{\bf{e}}^{2t}}\,\vec i + \left( {2 - {t^2}} \right)\vec j - {{\bf{e}}^{2t}}\vec k\) Solution

For problems 3 & 4 find the tangent line to the vector function at the given point.

  1. \(\vec r\left( t \right) = \cos \left( {4t} \right)\vec i + 3\sin \left( {4t} \right)\vec j + {t^3}\vec k\) at \(t = \pi \). Solution
  2. \(\displaystyle \vec r\left( t \right) = \left\langle {7{{\bf{e}}^{2 - t}},\frac{{16}}{{{t^3}}},5 - t} \right\rangle \) at \(t = 2\). Solution
  3. Find the unit normal and the binormal vectors for the following vector function. \(\vec r\left( t \right) = \left\langle {\cos \left( {2t} \right),\sin \left( {2t} \right),3} \right\rangle \) Solution