Alignment using an alignat{2} environment

Question

In the following code, I have the appropriate alignment. All the equal signs are aligned. There is one expression that is written on two lines; I wanted the first plus signs aligned. I used an aligned[t] environment to get this. How do I get this using an alignat{2} environment?

\documentclass[draft,a4paper,landscape]{amsart}
\usepackage{}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{newlfont}
\usepackage{mathtools,array}

\begin{document}

\noindent $a = x_{1} + iy_{1}$ and $z = x_{2} + iy_{2}$.
\begin{align*}
\big\vert 1 - \overline{a}z \big\vert^{2} &= \big\vert 1 - x_{1}x_{2} - y_{1}y_{2} + i(x_{1}y_{2} - x_{2}y_{1}) \big\vert^{2} \\
&= \begin{aligned}[t]1 &+ {x_{1}}^{2}{x_{2}}^{2} + {y_{1}}^{2}{y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} + 2x_{1}x_{2}y_{1}y_{2} \\
&+ {x_{1}}^{2}{y_{2}}^{2} + {x_{2}}^{2}{y_{1}}^{2} - 2x_{1}x_{2}y_{1}y_{2}
\end{aligned} \\
&= \begin{aligned}[t]1 + {x_{1}}^{2}{x_{2}}^{2} + {y_{1}}^{2}{y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} + {x_{1}}^{2}{y_{2}}^{2} + {x_{2}}^{2}{y_{1}}^{2}
\end{aligned} \\
&= \begin{aligned}[t]1 + ({x_{1}}^{2} + {y_{1}}^{2}) ({x_{2}}^{2} + {y_{2}}^{2}) - 2x_{1}x_{2} - 2y_{1}y_{2} ,
\end{aligned}
\intertext{and}
\vert z - a \vert^{2} &= \mathmakebox[0pt][l]{\vert x_{2} - x_{1} + i(y_{2} - y_{1}) \vert^{2}} \\
&= {x_{1}}^{2} + {x_{2}}^{2} + {y_{1}}^{2} + {y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} .
\end{align*}

\end{document}

Answer 1

I think it would be much too complicated to use alignat. Actually you only need one aligned environment. Note the \! to have an exact (first) alignment.

I don't see the necessity of aligning the + signs of the second and third lines of the alignment. I prefer aligning differently, so as to show clearly the third line is the continuation of the second ; I also made a smaller vertical spacing between these lines, thus making, in my opinion, the equations more readable.

If you want to use alignat it will be more complex — it wil be alignat{3} and a number of \rlap commands:

\documentclass[draft,a4paper,landscape]{amsart}
\usepackage{}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{newlfont}
\usepackage{mathtools,array}

\begin{document}

\noindent $a = x_{1} + iy_{1}$ and $z = x_{2} + iy_{2}$.
\begin{align*}
\big\vert 1 - \overline{a}z \big\vert^{2} &= \big\vert 1 - x_{1}x_{2} - y_{1}y_{2} + i(x_{1}y_{2} - x_{2}y_{1}) \big\vert^{2} \\
&=\!\begin{aligned}[t]1 + x_{1}^{2}x_{2}^{2}+ y_{1}^{2}y_{2}^{2} &- 2x_{1}x_{2} - 2y_{1}y_{2} + 2x_{1}x_{2}y_{1}y_{2} \\[-0.5ex]
  & + x_{1}^{2}y_{2}^{2} + x_{2}^{2}y_{1}^{2} - 2x_{1}x_{2}y_{1}y_{2}
 \end{aligned}
 \\
&=1  + x_{1}^{2}x_{2}^{2} + y_{1}^{2}y_{2}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} + x_{1}^{2}y_{2}^{2} + x_{2}^{2}y_{1}^{2}
 \\
&= 1 + (x_{1}^{2} + y_{1}^{2}) (x_{2}^{2} + y_{2}^{2}) - 2x_{1}x_{2} - 2y_{1}y_{2} ,
\intertext{and}
\vert z - a \vert^{2} &= \mathmakebox[0pt][l]{\vert x_{2} - x_{1} + i(y_{2} - y_{1}) \vert^{2}} \\
&= x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} .
\end{align*}\vskip1cm

With an \texttt{alignat} environment,  it is much more complex: 
\begin{alignat*}{3}
\big\lvert 1 - \overline{a}z \big\rvert^{2} &= \mathrlap{\big\lvert 1   - x_{1}x_{2} - y_{1}y_{2}{}+ i(x_{1}y_{2} - x_{2}y_{1}) \big\rvert^{2}} \\
&= 1& &{}  + x_{1}^{2}x_{2}^{2}+ y_{1}^{2} y_{2}^{2}  &  &{} - 2x_{1}x_{2} - 2y_{1}y_{2} + 2x_{1}x_{2}y_{1}y_{2} \\[-0.5ex]
  & &  & &  &    + x_{1}^{2}y_{2}^{2} + x_{2}^{2}y_{1}^{2} - 2x_{1}x_{2}y_{1}y_{2}
 \\
&=1 &&{} + \mathrlap{x_{1}^{2}x_{2}^{2} + y_{1}^{2}y_{2}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} + x_{1}^{2}y_{2}^{2} + x_{2}^{2}y_{1}^{2}}
 \\
&=1& &{}+ \mathrlap{(x_{1}^{2} + y_{1}^{2}) (x_{2}^{2} + y_{2}^{2}) - 2x_{1}x_{2} - 2y_{1}y_{2} ,}
\intertext{and}
\vert z - a \vert^{2} &= \mathrlap{\vert x_{2} - x_{1} + i(y_{2} - y_{1}) \vert^{2}} \\
&=\mathrlap{ x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2}.}
\end{alignat*}

\end{document} 

Answer 2

Here is one way to use the alignat environment:

Notes:

• Replaced the \intertext with \shortintertext as I think that looks better when you have very small text as it adds less vertical space before and after the text.
• Used \mathrlap to ensure that portions of some lines did not affect the alignment in the other lines.
• Aligned as many of the binary operators as I thought made sense without reorganizing the terms of the equations, or that did not leave too much blank space. Adjust this as desired.
• Since _each & provides a r and l alignment point, double && are used for all subsequent alignment points so that the text following the && is left aligned.

References:

• Can I use \clap, \rlap and \llap in math mode? especially the link in one of the answers: A complement to \smash, \llap, and \rlap

Code:

\documentclass[draft,a4paper,landscape]{amsart}
\usepackage{}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{newlfont}
\usepackage{mathtools,array}
\begin{document}
\noindent $a = x_{1} + iy_{1}$ and $z = x_{2} + iy_{2}$.
\begin{alignat}{7}
\big\vert 1 - \overline{a}z \big\vert^{2} 
&= \big\vert &&1 &&- x_{1}x_{2} - y_{1}y_{2} + \mathrlap{i(x_{1}y_{2} - x_{2}y_{1}) \big\vert^{2}} \
&= && 1 &&+ {x_{1}}^{2}{x_{2}}^{2} + {y_{1}}^{2}{y_{2}}^{2} - 2x_{1}x_{2}  &&- 2y_{1}y_{2}  &&+ 2x_{1}x_{2}y_{1}y_{2} \
& && &&+ {x_{1}}^{2}{y_{2}}^{2} + {x_{2}}^{2}{y_{1}}^{2}  && &&- 2x_{1}x_{2}y_{1}y_{2}  \
&= && 1 &&+ {x_{1}}^{2}{x_{2}}^{2} + {y_{1}}^{2}{y_{2}}^{2} - 2x_{1}x_{2}  &&- 2y_{1}y_{2} &&+ {x_{1}}^{2}{y_{2}}^{2} + {x_{2}}^{2}{y_{1}}^{2} \
&= && 1 &&+ ({x_{1}}^{2} + {y_{1}}^{2}) ({x_{2}}^{2} + {y_{2}}^{2}) &&- 2x_{1}x_{2} &&- 2y_{1}y_{2} ,
\shortintertext{and}% <--- Replaced \intertext
\vert z - a \vert^{2} &= \mathrlap{\vert x_{2} - x_{1} + i(y_{2} - y_{1}) \vert^{2}} \
&= \mathrlap{{x_{1}}^{2} + {x_{2}}^{2} + {y_{1}}^{2} + {y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2}} .
\end{alignat}
\end{document}

Answer 3

Here is a solution which doesn't used an environment:

\documentclass[draft,a4paper,landscape]{amsart}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{newlfont}
\usepackage{mathtools}

\begin{document}

\noindent $a = x_{1} + iy_{1}$ and $z = x_{2} + iy_{2}$.
\begin{align*}
\big\vert 1 - \overline{a}z \big\vert^{2} &= \big\vert 1 - x_{1}x_{2} - y_{1}y_{2} + i(x_{1}y_{2} - x_{2}y_{1}) \big\vert^{2} \\
&= 1 + {x_{1}}^{2}{x_{2}}^{2} + {y_{1}}^{2}{y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} + 2x_{1}x_{2}y_{1}y_{2} \\
&\phantom{{}=1} + {x_{1}}^{2}{y_{2}}^{2} + {x_{2}}^{2}{y_{1}}^{2} - 2x_{1}x_{2}y_{1}y_{2}\\
&= 1 + {x_{1}}^{2}{x_{2}}^{2} + {y_{1}}^{2}{y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} + {x_{1}}^{2}{y_{2}}^{2} + {x_{2}}^{2}{y_{1}}^{2}\\
&= 1 + ({x_{1}}^{2} + {y_{1}}^{2}) ({x_{2}}^{2} + {y_{2}}^{2}) - 2x_{1}x_{2} - 2y_{1}y_{2} ,
\intertext{and}
\vert z - a \vert^{2} &= \vert x_{2} - x_{1} + i(y_{2} - y_{1}) \vert^{2} \\
&= {x_{1}}^{2} + {x_{2}}^{2} + {y_{1}}^{2} + {y_{2}}^{2} - 2x_{1}x_{2} - 2y_{1}y_{2} .
\end{align*}

\end{document}

You have a lot of parentheses which are not really needed. Here is my way:

\documentclass[draft,a4paper,landscape]{amsart}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{newlfont}
\usepackage{mathtools}

\begin{document}

\noindent $a = x_1 + iy_1$ and $z = x_2 + iy_2$.
\begin{align*}
\left| 1 - \overline{a}z \right|^2 &= \left| 1 - x_1x_2 - y_1y_2 + i\left(x_1y_2 - x_2y_1\right) \right|^2 \\
&= 1 + x_1^2x_2^2 + y_1^2y_2^2 - 2x_1x_2 - 2y_1y_2 + 2x_1x_2y_1y_2 \\
&\phantom{{}=1} + x_1^2y_2^2 + x_2^2y_1^2 - 2x_1x_2y_1y_2\\
&= 1 + x_1^2x_2^2 + y_1^2y_2^2 - 2x_1x_2 - 2y_1y_2 + x_1^2y_2^2 + x_2^2y_1^2\\
&= 1 + \left(x_1^2 + y_1^2\right) \left(x_2^2 + y_2^2\right) - 2x_1x_2 - 2y_1y_2 ,
\shortintertext{and}
\left| z - a \right|^2 &= \left| x_2 - x_1 + i\left(y_2 - y_1\right) \right|^2 \\
&= x_1^2 + x_2^2 + y_1^2 + y_2^2 - 2x_1x_2 - 2y_1y_2 .
\end{align*}

\end{document}