Find the vectors T, N, and B at the given point. r(t) =<t^2, \frac{2}{3}t^3 , t> and point <4,-\frac{16}{3},-2>

Zoe Oneal 2021-06-01 Answered
Find the vectors T, N, and B at the given point.
r(t)=<t2,23t3,t> and point <4,163,2>
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Expert Answer

smallq9
Answered 2021-06-02 Author has 106 answers
Jeffrey Jordon
Answered 2021-09-07 Author has 2262 answers

Given R(t)=<t2,23t3,t> and point <4,163,2>

The point <4,163,2> occursat t=-2

Find the derivative of the vector,

R(t)=<2t,2t2,1>

|R(t)|=(2t)2+(2t2)2+12

=4t2+4t4+1

=(2t2+1)2

=2t2+1

Tangent vectors:

T(t)=R(t)|R(t)|

=12t2+1<2t,2t2,1>

T(2)=12(2)2+1<2(2),2(2)2,1>

T(2)=12(2)2+1<2(2),2(2)2,1>

=<49,89,19>

T(t)=<(2t2+1)22t(4t)(2t2+1)2,(2t2+1)4t(2t2)(4t)(2t2+1)2,4t(2t2+1)2>

=<4t2+28t2(2t2+1)2,8t3+4t8t3(2t2+1)2,4t(2t2+1)2)>

=<24t2(2t2+1)2,4t(2t2+1)2,4t(2t2+1)2>

|T(t)|=(24t2)2+(4t)2+(4t)2(2t2+1)4

=1(2t2+1)2416t2+16t4+16t2+16t2

=1(2t2+1)216t4+16t2+4

=2(2t2+1)(2t2+1)2

=22t2+1

The normal vectors.

N(t)=T(t)|T(t)|=
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