Vector Functions

Question

r(t) = < e^5tcos(12t), e^5tsin(12t), e^5t >

 

1. Find the unit binomial vector B(t)

 

2. Find the curvature k(t)

 

3. Find the torsion 

 

4. Find the acceleration

 

5. Find the tangential component a_{T =} a(t)

 

6. Find the normal component a_N = a(t)

 

7. Verify by direct substitution that a = a_TT + a_NN 

Answer 1

\documentclass{article}

\usepackage{amsmath}

\usepackage{amsfonts}

\begin{document}

\title{Vector Calculus Solution}

\author{}

\date{}

\maketitle

\textbf{Given}

\[

r(t)=\langle e^{5t}\cos(12t),\; e^{5t}\sin(12t),\; e^{5t}\rangle

\]

\section*{1. Unit Binormal Vector}

\[

B(t)=\frac{r'(t)\times r''(t)}{|r'(t)\times r''(t)|}

\]

\[

B(t)=\frac{1}{\sqrt{194}}

\langle

12\sin(12t)-5\cos(12t),

-12\cos(12t)-5\sin(12t),

12

\rangle

\]

\section*{2. Curvature}

\[

k(t)=\frac{|r'(t)\times r''(t)|}{|r'(t)|^3}

\]

\[

k(t)=\frac{12}{194e^{5t}}

\]

\section*{3. Torsion}

\[

\tau=\frac{(r'\times r'')\cdot r'''}{|r'\times r''|^2}

\]

\[

\tau=\frac{5}{194}

\]

\section*{4. Acceleration}

\[

a(t)=r''(t)

\]

\[

a(t)=

\langle

e^{5t}(-119\cos12t-120\sin12t),

e^{5t}(-119\sin12t+120\cos12t),

25e^{5t}

\rangle

\]

\section*{5. Tangential Component}

\[

a_T=\frac{d}{dt}|v|

\]

\[

a_T=5\sqrt{194}\,e^{5t}

\]

\section*{6. Normal Component}

\[

a_N=\frac{|v\times a|}{|v|}

\]

\[

a_N=12\sqrt{194}\,e^{5t}

\]

\section*{7. Verification}

\[

a=a_TT+a_NN

\]

\[

a=

(5\sqrt{194}e^{5t})T+(12\sqrt{194}e^{5t})N

\]

\[

a=r''(t)

\]

Hence verified.

Just put this code in latex r overleaf and get ur solutions

\end{document}