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\documentclass[]{article}
%opening
\title{This is my homework}
\author{Ryan}
\date{}
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{braket}
\usepackage[margin=1.0in]{geometry}
\newcommand{\ehx}{\hat{e}_x}
\newcommand{\ehy}{\hat{e}_y}
\newcommand{\ehz}{\hat{e}_z}
\newcommand{\bd}{\textbf}
\begin{document}
\maketitle
\section{Problem 1}
Find the x component of the following vector
\begin{equation}
\ket{v}=a\ehx+b\ehy+c\ehz
\end{equation}
\bd{Soloution}\\
The x component of $\ket{v}$ is just the inner product of $\ker{v}$ vector with the x basis vector
\begin{equation}
\begin{split}
\braket{x|v}&=a\ehx\cdot\ehx+b\ehy\cdot\ehy+c\ehz\cdot\ehz\\
&=a
\end{split}
\end{equation}
\section{Problem 2}
Prove the earth is Ricci flat
\begin{equation}
\begin{pmatrix}
1&0\\
0&1
\end{pmatrix}
\end{equation}
\end{document}
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