Calculating T(t) and N(t) from R(T) position vector

[katiek46]

Hi - I'm attempting to calculate the principal normal vector and tangent vector to R(t) and am having some problems. Any help would be appreciated!

R(t) = <t^2, t^3>
Ive got R`(t) as <2t,3t^2> and llR`(t)ll as t(4+9t^2)^1/2
so T(t) = <2/(4+9t^2)^1/2 , 3t/(4+9t^2)^1/2

My problem arises when attempting to calculate N(t)
I know that N(t) = T`(t) / llT`(t)ll
What is the easiest way to calculate these? Are there tricks?
I've no problem differentiating i which I've got as -18t/(4+9t^2)^3/2 but am having problems figuring j which I'm attempting to do using chain rule....
I've gotten as far as 3/(4+9t^2)^1/2 -9/(4+9t^2)^3/2.
HELP!

Can you also indicate the easiest way to determine the llT`(t) ll?

Thank you so much! Katie

 

[pka]

You know that \(\displaystyle \L
T = \frac{{R'}}{{\left\| {R'} \right\|}}\).

From that we can show that \(\displaystyle \L
T' = \frac{{R' \times (R'' \times R')}}{{\left\| {R'} \right\|^3 }}\).

Thus \(\displaystyle \L
N = \frac{{T'}}{{\left\| {T'} \right\|}} = \frac{{R' \times (R'' \times R')}}{{\left\| {R' \times (R'' \times R')} \right\|}}\).

If you don’t mind using cross products of functions, this does give you a way to find N from R, R’ & R”.
I remind you that we derive these using: \(\displaystyle \L
A \times (B \times C) = (A \cdot C)B - (A \cdot B)C\).

 

[daon]

I remember these problems. My professor gave us one on an exam that took up 2 pages of scribbling to figure out. Bad memories :evil: