If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
-Find the velocity vector.
$\mathbf { r } ( \mathrm { t } ) = \left( - 8 \mathrm { t } ^ { 2 } - 3 \right) \mathbf { i } + \left( \frac { 1 } { 9 } \mathrm { t } ^ { 3 } \right) \mathbf { j }$
A) $v = ( - 16 ) i + \left( \frac { 2 } { 3 } t \right) j$
B) $\mathbf { v } = ( - 16 \mathrm { t } ) \mathbf { i } + \left( \frac { 1 } { 3 } \mathrm { t } ^ { 2 } \right) j$
C) $\mathbf { v } = ( - 16 \mathrm { t } ) \mathbf { i } - \left( \frac { 1 } { 3 } \mathrm { t } ^ { 2 } \right) \mathbf { j }$
D) $\mathbf { v } = \left( \frac { 1 } { 3 } \mathrm { t } ^ { 2 } \right) \mathbf { i } + ( - 16 \mathrm { t } ) \mathbf { j }$

Question

Quizplus · Accepted Answer

The velocity vector $\mathbf{v}(t)$ is found by differentiating the position vector $\mathbf{r}(t)$ with respect to time $t$. Differentiating each component of $\mathbf{r}(t) = (-8t^2 - 3)\mathbf{i} + \left(\frac{1}{9}t^3\right)\mathbf{j}$ gives $\mathbf{v}(t) = (-16t)\mathbf{i} + \left(\frac{1}{3}t^2\right)\mathbf{j}$, which matches choice B.

If R(t) Is the Position Vector of a Particle in the Plane