Question

Help with some vector stuff (Calc III)

Answer 1

\[r(t)=3e^t<\cos(t),\sin(t),1>\]
\[r'(t)=3e^t<-\sin(t),\cos(t),0>\]
\[r''(t)=3e^t<-\cos(t),-\sin(t),0>\]
\[||r't||=3e^t\]

Answer 2

Doing dot product with this 3e^t hanging aroung is really throwing me off

Answer 3

$$ \vec{a} = {a_T} \vec{T} + a_N \vec{N} $$

Answer 4

what do you mean by that

Answer 5

$$ \large \rm {\vec{a} = {a_T} \vec{T} + a_N \vec{N}
\\ \, \\
a_T = v' = \frac{\vec{ r}'(t) \cdot \vec{r} '' (t) }{\|r'(t) \|}
\\ \, \\
a_N = \frac{ \| \vec{ r}'(t) \times \vec{ r}''(t) \| } { \| \vec{ r}'(t) \| }

}$$

Answer 6

These equations are supplied here.
http://tutorial.math.lamar.edu/Classes/CalcIII/Velocity_Acceleration.aspx

Answer 7

$$a_T = \| \vec{v}(t) \|' = \frac{\vec{ r}'(t) \cdot \vec{r} '' (t) }{\|r'(t) \|}

\\ \, \\
a_T = \frac{19e^{2t} }{ \sqrt{19} e^{t} } $$

Answer 8

I got it, thank you @owen3