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Section 12.8 : Tangent, Normal and Binormal Vectors

For problems 1 – 3 find the unit tangent vector for the given vector function.

1. \(\vec r\left( t \right) = {t^2}\,\vec i - \cos \left( {8t} \right)\vec j + \sin \left( {8t} \right)\vec k\)
2. \(\vec r\left( t \right) = \left\langle {8t,2 - {t^6},{t^4}} \right\rangle \)
3. \(\vec r\left( t \right) = \left\langle {\ln \left( {6t} \right),{{\bf{e}}^{1 - t}},5t} \right\rangle \)

For problems 4 & 5 find the tangent line to the vector function at the given point.

1. \(\vec r\left( t \right) = \left\langle {3 + {t^2},{t^4},6} \right\rangle \) at \(t = - 1\).
2. \(\vec r\left( t \right) = \left\langle {2t,{{\cos }^2}\left( t \right),{{\bf{e}}^{6t}}} \right\rangle \) at \(t = 0\).

For problems 6 & 7 find the unit normal and the binormal vectors for the given vector function.

1. \(\vec r\left( t \right) = \left\langle {{{\bf{e}}^{4t}}\sin \left( t \right),{{\bf{e}}^{4t}}\cos \left( t \right),2} \right\rangle \)
2. \(\vec r\left( t \right) = 2t\,\vec i + \frac{1}{2}{t^2}\,\vec j + \ln \left( {{t^2}} \right)\vec k\)