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\noindent \underline{Homework 5 -- MATH 1586 Spring 2018} \\
Recall that to add vectors, 
\[ \langle a,b,c \rangle + \langle d,e,f \rangle = \langle a+d, b+e, c+f \rangle.\]
For a real number $\alpha$, scalar multiplication obeys
\[ \alpha \langle a,b,c \rangle = \langle \alpha a, \alpha b, \alpha c \rangle.\]
The dot product is given by
\[ \langle a,b,c \rangle \cdot \langle d,e,f \rangle = ad+be+cf.\]
Recall the determinant of a $2 \times 2$ (i.e. two rows, two columns) is
\[\mathrm{det} \begin{bmatrix} a&b \\ c&d \end{bmatrix} = ad-bc.\]
Recall the determinant of a $3 \times 3$ matrix can be computers
\[ \mathrm{det} \begin{bmatrix} a&b&c \\ d&e&f \\ g&h&i \end{bmatrix} = a \begin{bmatrix} e&f \\ h&i \end{bmatrix} - b \begin{bmatrix} d&f \\ g&i \end{bmatrix} + c \begin{bmatrix} d&e \\ g&h \end{bmatrix}.\]
Also recall the ``basis vectors" $\vec{i} = \langle 1,0,0 \rangle$, $\vec{j} = \langle 0,1,0 \rangle$, and $\vec{k} = \langle 0,0,1 \rangle$.
The cross product of two vectors is given by
\[ \langle a,b,c \rangle \times \langle d,e,f \rangle = \det \begin{bmatrix} \vec{i} & \vec{j} & \vec{k} \\ a&b&c \\ d&e&f \end{bmatrix}.\]
Recall that differentiation and integration work componentwise: if $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$, then
\[ \dfrac{\mathrm{d}}{\mathrm{d}t} \vec{r}(t) = \langle f'(t), g'(t), h'(t) \rangle, \]
and
\[ \displaystyle\int_a^b \vec{r}(t) \mathrm{d}t = \left\langle \displaystyle\int_a^b f(t) \mathrm{d}t, \displaystyle\int_a^b g(t) \mathrm{d}t, \displaystyle\int_a^b h(t) \mathrm{d}t \right\rangle.\]
\begin{enumerate}

\item Calculate $\langle 3,7,10 \rangle + 2 \langle 3,1,-1 \rangle$.

\item Compute the dot product $\langle 1,6,2 \rangle \cdot \langle 7,1,3 \rangle$. 

\item Compute the determinant $\det \begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix}$.

\item Compute the cross product $\langle 1,0,1 \rangle \times \langle 2,1,3\rangle$. 

\item Let $\vec{r}(t)=\langle t^2+1, 3t-2, \ln(t) \rangle$. Compute $\vec{r}^{\hspace{2pt}\prime}(2)$. 

\item Let $\vec{r}(t) = \langle t^2+2t, t^3+3, e^{2t} \rangle$. Compute $\displaystyle\int_1^2 \vec{r}(t) \mathrm{d}t$. 

\item Suppose that a particle has an initial position of $\vec{r}(0)=\langle 1,2,3 \rangle$, initial velocity $\vec{v}(0)=\langle 1,1,-1\rangle$, and obeys acceleration $\vec{a}(t)=\langle 0,2,5\rangle$. Find the position of the particle at time $t$. 
\end{enumerate}

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