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Unit Tangent Vector
Hello,
I was just wondering if my solution to the following question looks good, if so, do I set
r'(t) = 1 and solve for t? Can anyone help???
Finding the Unit Tangent Vector to the curve give:
r(t) = (6+10t^7)i+ (6+11t^7)j +(6+2t^7)k
r ' (t)= 70t^6i +77^6j +14t^6k
Unit Tagent Vector:
T(t)= r'(t) / ll r' (t) ll
= 70t^6i +77^6j +14t^6k / sqrt{ (70t^6i)^2 +(77^6j)^2 + (14t^6k)^2 }
=70t^6i +77^6j +14t^6k / sqrt { 4900t^12 i + 5929t^12 + 196t^12) }
* Now I am pretty sure I solve for t if my arithmetic is correct!?!?!
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First, when finding the magnitude of a vector, such as  ||)
, don't square the unit vectors 
, 
, or 
, square only the components.
Regarding your actual question, you don't need to solve for 
. Like the position vector, the unit tangent vector varies from time to time, so naturally it is a function of time. Therefore, your answer of 
is correct (except for the "i" you placed after the 
in the denominator, which I removed).
Of course, your answer could be simplified. For example, the denominator is equal to 
. Assuming is positive, then  = \dfrac{2}{3} \vec i + \dfrac{11}{15} \vec j + \dfrac{2}{15} \vec k)
.
