\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{enumerate}
\begin{document}
\flushleft\underline{Quiz 4 --- MATH 1586 Spring 2018} \\
\begin{enumerate}
\item Calculate $\langle 1,2,3 \rangle + 2 \langle 1,-1,5 \rangle$. \\
\textit{Solution}: Calculate
$$\begin{array}{ll}
\langle 1,2,3\rangle + 2\langle 1,-1,5 \rangle &= \langle 1,2,3 \rangle + \langle 2,-2,10\rangle \\
&= \langle 1+2, 2-2, 3+10 \rangle \\
&=\langle 3, 0, 13 \rangle.
\end{array}$$
\item Calculate $\langle 1,2,3 \rangle \cdot \langle 1,-1,5\rangle$. \\
\textit{Solution}: Calculate
$$\begin{array}{ll}
\langle 1,2,3 \rangle \cdot \langle 1,-1,5 \rangle &= 1(1) + 2(-1) + 3(5) \\
&= 1-2+15 \\
&=14.
\end{array}$$

\item Calculate $\langle 1,2,3 \rangle \times \langle 1,-1,5 \rangle$. \\
\textit{Solution}: Calculate
$$\begin{array}{ll}
\langle 1,2,3 \rangle \times \langle 1,-1,5 \rangle &= \det \begin{bmatrix} \vec{i} & \vec{j} & \vec{k} \\
1&2&3 \\
1&-1&5 
\end{bmatrix} \\
&= \vec{i} \det \begin{bmatrix} 2&3 \\ -1&5 \end{bmatrix} - \vec{j} \det \begin{bmatrix} 1&3 \\ 1&5 \end{bmatrix} + \vec{k} \begin{bmatrix} 1&2 \\ 1&-1 \end{bmatrix} \\
&=\vec{i} (10-(-3)) - \vec{j} (5-3) + \vec{k} (-1 - 2) \\
&=13 \langle 1,0,0 \rangle - 2 \langle 0,1,0 \rangle -3 \langle 0,0,1 \rangle \\
&=\langle 13,-2,-3 \rangle.
\end{array}$$
\end{enumerate}
\end{document}