12.8-bo'lim : Tangent, normal va binormal vektorlar

1 – 3 masalalar uchun berilgan vektor funksiya uchun birlik tangens vektorini toping.

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. \(\vec r\left( t \right) = \left\langle {\ln \left( {6t} \right),{{\bf{e}}^{1 - t}},5t} \right\rangle \)

4 va 5-masalalar uchun berilgan nuqtadagi vektor funksiyasiga teginish chizig‘ini toping.

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

6 va 7-masalalar uchun berilgan vektor funksiya uchun normal birlikni va binormal vektorlarni toping.

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. \(\vec r\left( t \right) = 2t\,\vec i + \frac{1}{2}{t^2}\,\vec j + \ln \left( {{t^2}} \right)\vec k\)