Starting from the point \left(3,−2,−1\right), reparametrize the curve x

Suman Cole 2021-09-30 Answered
Starting from the point (3,2,1), reparametrize the curve x(t)=(33t,2t,1t) in terms of arclength.
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Expert Answer

Asma Vang
Answered 2021-10-01 Author has 93 answers

Step 1
First of all, the derivative of x is
x(t)=(3,1,1)
Its norm is
x(t)∣=(3)2+(1)2+(1)2=9+1+1=11
Now we find
s(t)=t0tx(u)du,
where x(t0) is the starting point So, we must have that
x(t0)=(3,2,1)
which yields
t0=0
Therefore,
s(t)=0tx(u)du=0t11du=11t
Therelore, we set
t=s11
This means that the reparametrization of x in terms of arclength is
x~(s)=x(s11)=(33s11,2s11,1s11)
To write it clearly,
x~(s)=(33s11,2s11,1s11)

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