Writing acceleration of a vector-valued function in terms of

[jaredld]

I am asked to write acceleration in the form of \(\displaystyle a = a_T T + a_N N\) at t=0 WITHOUT finding T and N. I will show you my thought on how this would be solved, however I am at a loss as to how to do it without finding T and N. Here goes nothing:
Given: \(\displaystyle r(t) = (2 + t) i\limits^ \wedge + (t + 2t^2 ) j\limits^ \wedge + (1 + t^2 ) k\limits^ \wedge\)
T=unit tangent vector, a=Acceleration, v=velocity;
then, \(\displaystyle T(t) = {{r'(t)} \over {\left\| {r'(t)} \right\|}} =\)\(\displaystyle {{v'(t)} \over {\left\| {v'(t)} \right\|}}\),
so if \(\displaystyle T(t) = {{v'(t)} \over {\left\| {v'(t)} \right\|}}\) then,
\(\displaystyle v = T*\left\| v \right\|\) additionally; \(\displaystyle a = v' = \left( {\frac{{d\left\| {v'} \right\|}}{{dt}} \times (t)} \right) + \left( {\left\| v \right\| \times \frac{{dT}}{{dt}}} \right)\)and in order to get it into
\(\displaystyle a = a_T\times T + a_N\timesN\) form, I multiply the right side by a form of one to obtain;
\(\displaystyle a = \frac{{d\left\| v ||}}
{{dt}} \times T(t) + (\left\| v \right\| \times \frac{{dT}}
{{dt}} \times \left\| {\frac{{\frac{{dT}}
{{dt}}}}
{{\frac{{dT}}
{{dt}}}}} \right\|) \to a = \frac{{d\left\| v \right\|}}
{{dt}} \times T(t) + (\left\| v \right\| \times \left\| {\frac{{dT}}
{{dt}}} \right\| \times N(t)\) the answer I have is \(\displaystyle a = 2\sqrt 2 \times T + 2\sqrt {10} N \to T = [\frac{1}
{{\sqrt 2 }}i + \frac{1}
{{\sqrt 2 }}j]\;and\;N = [\frac{2}
{{\sqrt 5 }}i + \frac{1}
{{\sqrt 5 }}k]\)
Does it seem correct? The problem before it had an answer like a=5T+10N...or something similar, so I am wondering if I am supposed to come up with some pretty constants out front? Am I thinking logically? Any help would be appreciated.

 

[pka]

You may be making too much out of this problem.
We can show each of the following:
\(\displaystyle \L
\begin{array}{l}
T = \frac{{R'}}{{\left\| {R'} \right\|}}\quad \& \quad N = \frac{{\left( {R' \cdot R'} \right)R'' - \left( {R' \cdot R''} \right)R'}}{{\left\| {\left( {R' \cdot R'} \right)R'' - \left( {R' \cdot R''} \right)R'} \right\|}} \\
a_T = \frac{{R' \cdot R''}}{{\left\| {R'} \right\|}}\quad \& \quad a_N = \frac{{\left( {\left\| {R' \times R''} \right\|} \right)}}{{\left\| {R'} \right\|}} \\
\end{array}\)

You can easily find each of these from the given:
\(\displaystyle \L
R'(0)\quad \& \quad R''(0)\).

The use the above to write \(\displaystyle \L
a(0) = a_T (0)T(0) + a_N (0)N(0)\)

 

[jaredld]

Yep I should have simplified everything down like you did in your post. It makes much more sense. My initial thought was that I should write everything out without missing a step so that people could follow my work easily and point out any mistakes in my thought process. I didn't realize at the time how convoluted my post would end up by doing it that way. Also I made some mistakes in writing code that I would have caught had I proofed it before posting. I apologize for that.

Thanks again for the help.

By the way, what do you guys do for work if you don't mind me asking? If you do just ignore me. I imagine a lot of you have to be professors or engineers or some such to know calculus this well. ......I bet most of you are professors aren't you? Because Engineers know calculus, but generally use trig most of the time for the work they have to do and so unless they keep in practice by helping people like me, they forget the stuff they don't use as much.