Question

Reparameterize the curve r(t) = <2t,1-3t, 5 + 4t> with respect to the arclength s, measured from the point t= 0.

Answer 1

$s = \int_0^t\vert\vert r'(v)\vert\vert \mathrm{d}v \\ r'(t) = 2\textbf{i} - 3\textbf{j} + 4\textbf{k}\\ r'(v) = 2\textbf{i} - 3\textbf{j} + 4\textbf{k}\\ \vert\vert r'(v) \vert\vert = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{29}\\ \therefore s = \int_0^t \sqrt{29} \mathrm{d}v = \sqrt{29}(t - 0) = \sqrt{29}t\\ \Rightarrow t = \frac{s}{\sqrt{29}} \\ \displaystyle\therefore r(t(s)) = \left(\frac{2s}{\sqrt{29}},1-\frac{3s}{\sqrt{29}}, 5 + \frac{4s}{\sqrt{29}}\right) \\\textsf{is the required reparameterized}\\\textsf{equation of the curve}$