Determine whether the parameter used in each vector function is arc length.

(a) $C_{1}$: $\ \mathbf{r}(t)=2\sin \dfrac{t}{2}\mathbf{i}+2\cos \dfrac{t}{2}\mathbf{j}\quad 0\leq t\leq 2\pi$

(b) $C_{2}$: $\ \mathbf{r}(t)=\cos t\mathbf{i}+\sin t\mathbf{j}+t\mathbf{k}\quad t\geq 0$

Solution (a) We begin by finding $\mathbf{r}^{\prime} ( t)$ and $\left\Vert \mathbf{r^{\prime} }( t) \right\Vert .$ $$\begin{equation*} \begin{array}{rrr} \mathbf{r}^{\prime} (t)=\cos \dfrac{t}{2}\mathbf{i}-\sin \dfrac{t}{2}\mathbf{j} \qquad \left\Vert \mathbf{r}^{\prime} (t)\right\Vert\;=\;\sqrt{ \cos ^{2}\dfrac{t}{2}+\sin ^{2}\dfrac{t}{2}}=1\quad \hbox{for all}~t \end{array} \end{equation*}$$

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Since $\left\Vert \mathbf{r}^{\prime} (t)\right\Vert\;=\;1$ for all $t,$ the parameter $t$ is arc length as measured along $C_{1}$.

(b) We begin by finding $\mathbf{r}^{\prime} ( t)$ and $\left\Vert \mathbf{r^{\prime} }( t) \right\Vert .$ $$\begin{equation*} \begin{array}{rrr} \mathbf{r}^{\prime} (t)=-\sin t\mathbf{i}+\cos t\mathbf{j+k}\qquad \left\Vert \mathbf{r}^{\prime} (t)\right\Vert\;=\;\sqrt{\sin ^{2}t+\cos ^{2}t+1}=\sqrt{2} \end{array} \end{equation*}$$

Since $\left\Vert \mathbf{r}^{\prime }(t)\right\Vert\;\neq 1$ for all $t$, the parameter $t$ does not measure arc length along $C_{2}$.