$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. $$\begin{array} { l }
\mathbf { r } ( t ) = 2 t \mathbf { i } - 3 t ^ { 3 } \mathbf { j } + 3 t ^ { 2 } \mathbf { k } \
\mathbf { u } ( t ) = 3 \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 4 t ^ { 3 } \mathbf { k }
\end{array}$$
A) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t ^ { 2 } - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$
B) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 32 t + 18 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$
C) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } - 3 t ^ { 2 } \mathbf { k }$
D) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$
E) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t + 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 3 } \mathbf { k }$

Question

$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. $$\begin{array} { l } 
\mathbf { r } ( t ) = 2 t \mathbf { i } - 3 t ^ { 3 } \mathbf { j } + 3 t ^ { 2 } \mathbf { k } \
\mathbf { u } ( t ) = 3 \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 4 t ^ { 3 } \mathbf { k }
\end{array}$$
A) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t ^ { 2 } - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$
B) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 32 t + 18 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$
C) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } - 3 t ^ { 2 } \mathbf { k }$
D) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$
E) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t + 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 3 } \mathbf { k }$

Quizplus · Accepted Answer

We use the following properties of the derivative:
1. $\dfrac{d}{dt}(c\mathbf{u}(t)) = c\dfrac{d}{dt}\mathbf{u}(t)$ where $c$ is a constant.
2. $\dfrac{d}{dt}\mathbf{u}(t)	imes\mathbf{v}(t) = \dfrac{d}{dt}\mathbf{u}(t)	imes\mathbf{v}(t) + \mathbf{u}(t)	imes\dfrac{d}{dt}\mathbf{v}(t)$
3. $\dfrac{d}{dt}(f(t)\mathbf{u}(t)) = f'(t)\mathbf{u}(t) + f(t)\dfrac{d}{dt}\mathbf{u}(t)$ where $f(t)$ is a scalar function.

Using property (1), we simplify $\mathbf{r}(t)	imes\mathbf{u}(t)$ as:

\begin{align*}
\mathbf{r}(t)	imes\mathbf{u}(t) &= (2t\mathbf{i}-3t^3\mathbf{j}+3t^2\mathbf{k})	imes(3\mathbf{i}+5t^2\mathbf{j}+4t^3\mathbf{k})\
&= 6t\mathbf{j}	imes\mathbf{i} - 10t^2\mathbf{k}	imes\mathbf{i} - 9t^3\mathbf{i}	imes\mathbf{j} - 15t^5\mathbf{k}	imes\mathbf{j} + 9t^4\mathbf{i}	imes\mathbf{k} - 12t^3\mathbf{j}	imes\mathbf{k}\
&= -10t^2\mathbf{j} - 9t^3\mathbf{k} + 15t^5\mathbf{i}
\end{align*}

Now using property (3), we find the derivative of the original expression:

\begin{align*}
\dfrac{d}{dt}(\mathbf{r}(t)	imes\mathbf{u}(t)) &= \dfrac{d}{dt}(-10t^2\mathbf{j} - 9t^3\mathbf{k} + 15t^5\mathbf{i})\
&= 0\cdot\mathbf{j} - 27t^2\mathbf{k} + 75t^4\mathbf{i}\
&= \boxed{\left(-60t^3 - 72t^5ight)\mathbf{i} + \left(18t-32t^3ight)\mathbf{j} + 57t^2\mathbf{k}}
\end{align*}

Dt[r(t)×u(t)] given the following D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }Dt​[r(t)×u(t)] given the following

$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$