Question

Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = 3t i + (7 − 4t) j + (3 + 2t) k

Answer 1

The curve is given by r(s)=
`3(s)/(√(29))  i + (7 - 4(s)/(√(29)) ) j + (3 + 2(s)/(√(29)) ) k`

Explanation:

The aec length of the curve is given by:

L=
`\int\limits^t_0 \, dt`

r(t) = 3t i + (7 − 4t) j + (3 + 2t) k

r'(t)= 3 i -4 j + 2 k

|r'(t)|=
`√(3^2+(-4)^2+2^2) =√(9+16+4) =√(29)`

L=
`\int\limits^t_0 {√(29)} \, dt=√(29)\int\limits^t_0 \, dt`

L=
`√(29)t|^t_0=√(29)t-√(29)0=√(29)t`

s(t)=
`√(29)t`

We have to obtain t in terms of s, hence:

t=
`(s)/(√(29))`

Finally,

r(s)=
`3(s)/(√(29))  i + (7 - 4(s)/(√(29)) ) j + (3 + 2(s)/(√(29)) ) k`