Unit tangent vector problem: Vector r(t) = for t is great than or equal to 1. Find unit tangent vector.

Question

Unit tangent vector problem:
Vector r(t) = <t, 2, 2/t> for t is great than or equal to 1. Find unit tangent vector.

anonymous · Answer

@ganeshie8

ganeshie8 · Answer

definition :   $\large  \frac{r'(t)}{|r'(t)|}$

ganeshie8 · Answer

to take the derivative of a vector function, 
simply take derivative of each component

anonymous · Answer

I think I am having trouble factoring the magnitude on bottom.

anonymous · Answer

I have T(t)  = <1, 0, -2/t^2> / ( 1 + 4/t^4 )

ganeshie8 · Answer

looks good

anonymous · Answer

sorry  T(t)  = <1, 0, -2/t^2> / ( 1 + 4/t^4 )^1/2

ganeshie8 · Answer

oh yah

anonymous · Answer

my book shows <t^2, 0, -2>  (t^4 + 4)^1/2  ?

ganeshie8 · Answer

thats right

ganeshie8 · Answer

$\large \frac{<1, 0, -2/t^2>}{\sqrt{1 + 4/t^4}} $ $\large \frac{<1, 0, -2/t^2>}{\sqrt{(t^4 + 4)/t^4}} $

ganeshie8 · Answer

$\large   \frac{<1, 0, -2/t^2>}{\sqrt{(t^4 + 4)}/t^2} $

ganeshie8 · Answer

multiply top and bottom wid $t^2$

ganeshie8 · Answer

$\large   \frac{t^2<1, 0, -2/t^2>}{\sqrt{(t^4 + 4)}} $

anonymous · Answer

so did you first factor out t^4 ? O couldnt quite tell.

ganeshie8 · Answer

we have below in the denominator :-
$\large  1 + \frac{4}{t^2}$

right ?

ganeshie8 · Answer

to add fractions, u need to have common-denominator

ganeshie8 · Answer

$\large  1 + \frac{4}{t^4}$

$\large  1\times\frac{t^4}{t^4} + \frac{4}{t^4}$

$\large  \frac{t^4}{t^4} + \frac{4}{t^4}$

anonymous · Answer

got it.

ganeshie8 · Answer

now that the denominators are same, we can add the numerators directly :

$\large  1 + \frac{4}{t^4}$

$\large  1\times\frac{t^4}{t^4} + \frac{4}{t^4}$

$\large  \frac{t^4}{t^4} + \frac{4}{t^4}$

$\large  \frac{t^4+4}{t^4} $

anonymous · Answer

I see whre your going with it (; Can I show you one more ting fro a different eq. you mght know how to simplify in a similiar way? 4sin^2(2x) + 16cos^2(2x)

ganeshie8 · Answer

thats easy, write 16 as 4 + 12

ganeshie8 · Answer

4sin^2(2x) + 16cos^2(2x)

4sin^2(2x) + 4cos^2(2x) + 12cos^2(2x) 

4(sin^2(2x) + cos^2(2x) )  +  12cos^2(2x) 

4(1) +  12cos^2(2x) 

4 +  12cos^2(2x)

ganeshie8 · Answer

i dont think it simplifies anymore

anonymous · Answer

Thats it and thank you. What time is it in India?

ganeshie8 · Answer

np :) 1:04 pm .... going for lunch.... cya 
must be very late over there. lol u should goto sleep :)

anonymous · Answer

Its 11 36 pm (; No class tomorrow. Have a good day.