\documentclass[12pt]{article}

\usepackage{amsmath}

\begin{document}

\begin{eqnarray*}
\begin{pmatrix}
\mathbf{a}  &
\mathbf{b}  &
\mathbf{c} 
\end{pmatrix}
& = &
\begin{pmatrix}
a_x & b_x & c_x \\
0 & b_y & c_y \\
0 & 0 & c_z \\
\end{pmatrix} \\
a_x &=& A \\
b_x &=& \mathbf{B} \cdot \mathbf{\hat{A}} \quad = \quad B \cos{\gamma} \\
b_y &=& |\mathbf{\hat{A}} \times \mathbf{B}| \quad = \quad B \sin{\gamma} \quad =  \quad  \sqrt{B^2 - {b_x}^2} \\
c_x &=& \mathbf{C} \cdot \mathbf{\hat{A}} \quad = \quad C \cos{\beta} \\
c_y &=& \mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})} \times \mathbf{\hat{A}} \quad = \quad \frac{\mathbf{B} \cdot \mathbf{C} - b_x c_x}{b_y} \\
c_z &=& |\mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})}|\quad = \quad \sqrt{C^2 - {c_x}^2 - {c_y}^2} \\
\end{eqnarray*}

\end{document}