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### Section 1-8 : 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