Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$.
$\mathbf { r } ( t ) = ( 7 + 3 t ) \mathbf { i } + ( 10 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$
A) $\mathbf { r } ( t ( s ) ) = \left( 7 - \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$
B) $\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 - \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$
C) $\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$
D)
$\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } + ( 6 s ) \mathbf { k }$
E)
$\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - ( 6 s ) \mathbf { k }$

Question

Quizplus · Accepted Answer

To reparametrize the curve with respect to arc length from the point where $t = 0$, we first find the speed $v$ of the curve. The derivative of $\mathbf{r}(t)$ with respect to $t$ gives the velocity vector $\mathbf{r}'(t) = 3\mathbf{i} + 9\mathbf{j} - 6\mathbf{k}$. The speed is the magnitude of this velocity vector, $v = \sqrt{3^2 + 9^2 + (-6)^2} = \sqrt{9 + 81 + 36} = \sqrt{126}$. The arc length $s$ from the point where $t = 0$ is given by $s = vt = \sqrt{126}t$. Solving for $t$ gives $t = \frac{s}{\sqrt{126}}$. Substituting $t$ in terms of $s$ into the original parametrization gives the reparametrized curve with respect to arc length as $\mathbf{r}(t(s)) = (7 + \frac{3}{\sqrt{126}}s)\mathbf{i} + (10 + \frac{9}{\sqrt{126}}s)\mathbf{j} - (\frac{6s}{\sqrt{126}})\mathbf{k}$, which matches option C.

Reparametrize the Curve with Respect to Arc Length Measured from the Point