Make line extend beyond endpoint and intersect with line

Question

Is it possible to have a line go through two coordinates and then extend until it intersects with another line? The line in question is not horizontal nor vertical (otherwise I could use the perpendicular coordinate system).

My current code is

\begin{tikzpicture}

\coordinate[label={left:$(x, y, z)$}] (w) at (-6,2);
\coordinate (c) at (0,0);
\coordinate[label={right:$(x,y)$}] (b) at (3,-1);
% I want to avoid hard-coding this coordinate

\draw[ultra thick,-stealth] (w |- c) -- (w);
\draw[ultra thick,-stealth] (c -| b) -- (b);

\draw   (c) +(0,1.5) -- ++(0,-1.5);

\draw[dashed] (w |- c) -- (c -| b);
\draw[dashed] (w) -- (c); % Currently two paths to emulate the result
\draw[dashed] (c) -- (b);

\end{tikzpicture}

So, what I'd like is to specify a path that goes through point w and c and extends until it intersects with the (imaginary) vertical line at (3,y). I don't want to hardcode the position of point b, but have tikz calculate it for me.

Answer 1

You can solve a (very simple) linear equation with help of \pgfmathsetmacro and calc library:

The code:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
  % initial points: a, b, c
  \fill
  (-6,2) coordinate (a) circle(1pt) node[above]{a}
  (1,-1) coordinate (b) circle(1pt) node[above right]{b}
  (3,0)  coordinate (c) circle(1pt) node[above right]{c};

  % d intersects the line through (a) and (b) and the vertical line through (c)
  \path let \p1=(a), \p2=(b), \p3=(c) in
  \pgfextra{\pgfmathsetmacro{\y}{(\y1-\y2)/(\x1-\x2)*(\x3-\x2)+\y2}}
   (c |- 0,\y pt) coordinate (d) node[below]{d};

  \draw[ultra thick,-stealth] (a |- b) -- (a);
  \draw[ultra thick,-stealth] (b -| d) -- (d);
  \draw   (b) +(0,1.5) -- ++(0,-1.5);
  \draw[dashed] (a |- b) -- (b -| d);
  \draw[dashed] (a) -- (d);
\end{tikzpicture}
\end{document}

Answer 2

You can also use a to path though I didn't test it throughly so It is possible that this breaks down in some very sharp angles or the target is impossible to reach through given coordinate for example, through coordinate is on the left while the target is on the right with respect to the given starting point etc.

\documentclass[]{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\tikzset{
    passthrough/.style={
        to path={ let \p1=($ #1 - (\tikztostart)$),
        \p2=($ (\tikztotarget) - #1 $),
        \n1={atan2(\x1,\y1)},
        \n2={veclen(\x1,\y1) + veclen(\x2,0)/cos(\n1)}
        in (\tikztostart) -- ++(\n1:\n2)
        },
    }
}

\begin{document}
\begin{tikzpicture}

\coordinate[label={left:$(x, y, z)$}] (w) at (-6,2);
\coordinate (c) at (0,0);
\draw   (c) ++(0,1.5) -- ++(0,-3);

\draw[ultra thick,-stealth] (w |- c) -- (w);



\draw[line width=2mm,red] (w) to[passthrough=(c)] (4,0);
\draw[line width=1mm,blue] (w) to[passthrough=(c)] (3,0);
\draw[dashed] (-6,0) -- (3,0);     % Horizontal Test line for blue
\draw[dashed] (-6,-1) -- (4,-1);   % Horizontal Test line for red


\end{tikzpicture}
\end{document}