How do you find the unit tanget vector to the curve ?

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Homework Help Overview

The discussion revolves around finding the unit tangent vector and unit normal vector for the curve defined by the parametric equation r(t) = . The original poster is uncertain about how to proceed without a specific value for t.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the calculation of the tangent vector r'(t) and the unit tangent vector. There is a question about how to derive these vectors without a specific t value, leading to a discussion about generalization in terms of t. The original poster also seeks guidance on finding the unit normal vector from the unit tangent vector.

Discussion Status

Some participants have provided clarifications regarding the expressions for the tangent and normal vectors. There is an ongoing exploration of the correct notation and methods for differentiation, with no explicit consensus reached on the approach to take.

Contextual Notes

Participants note that the answers will remain in terms of the variable t, which is a common practice in calculus when specific values are not provided. There is mention of different methods for finding the normal vector, indicating varying levels of calculus knowledge among participants.

lexie04
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Homework Statement


r(t) = <cosh(t), -sinh(t), t>
not given a value for t.


Homework Equations



r'(t)
N(t)

The Attempt at a Solution



T = r'(t)/distance[r'(t)]

how do you find unit tangent vector and the unit normal vector without being given a value for t?

r'(t) = <sinh(t),- cosh(t),t>
 
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your r'(t) is not right. The tangent vector is r'(t), an the unit tangent vector is r'(t)/||r'(t)||
 
oh and to answer your question, the answers are in terms of t. since you are not given a value of t(it is a variable so there is no set value), your answer is in terms of t. this allows for the answer that you get to be 'generalized'. this means that you can substitute any value of t you want into your equation and get the answer. it looks to me like you will be dealing with these t variables exclusively for the rest of your calculus career
 
so when i get the unit tangent vector it is <sinh(t)/(sinh(t)+cosh(t)),-cosh(t)/(sinh(t)+cosh(t)),0> how do i go about taking the derivative of that in order to find the unit normal vector?
 
i believe the derivative of cosh(x) is sinh(x) and vice versa, you should double check that. and what level of calculus are you at? there are different methods for finding the normal vector and i want to know which one will suit your level. this is mostly because i can only remember the higher level one because its more recent in my memory.
 
calc 3. the derivatives where given to me. and the equation we have been using for N(t) is T'/dist(T').
thank you
 
and using that formula you are still having issues? I don't quite understand that notation. is dist(T') = ||T'||? or the magnitude of T'
 
It's supposed to be the magnitude of T'. T' is just the derivative of the tangent vector.
 

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