12.8-bo'lim : Tangent, normal va binormal vektorlar
3. \(t = \pi \) da \(\vec r\left( t \right) = \cos \left( {4t} \right)\vec i + 3\sin \left( {4t} \right)\vec j + {t^3}\vec k\) ga teguvchi chiziqni toping.
Yechim
Birinchidan, vektor funktsiyasiga teginish vektorini olishimiz kerak. Tangens vektori,
\[\vec r'\left( t \right) = - 4\sin \left( {4t} \right)\vec i + 12\cos \left( {4t} \right)\vec j + 3{t^2}\vec k\]E'tibor bering, agar biz xohlasak, bu erda birlik tangens vektoridan foydalanishimiz mumkin, ammo ular qanchalik chalkash bo'lishini hisobga olsak, biz buni amalga oshiramiz.
2-qadam
Endi bizga "to'liq" tangens vektori emas, balki muammo bayonotida berilgan \(t\) qiymatidagi tangens vektori kerak. Bizga muammo bayonidagi \(t\) qiymatidagi vektor funksiyasidagi nuqta ham kerak bo'ladi. Bular,
\[\begin{align*}\vec r\left( \pi \right) & = \cos \left( {4\pi } \right)\vec i + 3\sin \left( {4\pi } \right)\vec j + {\pi ^3}\vec k = \vec i + {\pi ^3}\vec k\\ \vec r'\left( \pi \right) & = - 4\sin \left( {4\pi } \right)\vec i + 12\cos \left( {4\pi } \right)\vec j + 3{\pi ^2}\vec k = 12\vec j + 3{\pi ^2}\vec k\end{align*}\]3-qadam
Tangens chiziq tenglamasini yozish uchun bizga chiziq ustidagi nuqta va chiziqqa parallel vektor kerak. Albatta, bu faqat oldingi bosqichda topilgan miqdorlar.
Tangens chiziq u holda,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\vec r\left( t \right) = \vec i + {\pi ^3}\vec k + t\left( {12\vec j + 3{\pi ^2}\vec k} \right) = \vec i + 12t\,\vec j + \left( {{\pi ^3} + 3{\pi ^2}t} \right)\vec k}}\]