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	\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}