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I am using the documentclass book in this example.

\documentclass{book}
\begin{document}
\chapter*{Test}
Lorem ipsum dolor sit amet...
\vspace{3cm}
Lorem ipsum dolor sit amet...
\end{document}

Now I want to know the distance between the title of a chapter and the first line of the text.

Output of the example above

The space between the two lines of dummy text (b) should be the same as the space between the chapter's title and the first line (a). There might be a default lenght which I can use instead of the 3cm which I set manually.

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Using \vspace{40pt} instead of \vspace{3cm} should give you the desired result:

enter image description here

You can find this length by looking up the definition of makeschapterhead in the file book.cls. There you can find the following lines:

def\@makeschapterhead#1{%
  \vspace*{50\p@}%
  {\parindent \z@ \raggedright
    \normalfont
    \interlinepenalty\@M
    \Huge \bfseries  #1\par\nobreak
    \vskip 40\p@
  }}

The important part in this case is \vskip 40\p@. \p@ represents 1pt (for an explanation see here: https://tex.stackexchange.com/a/38666/134144), so that we can use \vspace{40pt} in the document to get the same distance.

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0

Please add below macro on your preamble part (i.e., before \begin{document})

\makeatletter
\def\@makeschapterhead#1{%
  \vspace*{50\p@}%
  {\parindent \z@ \raggedright
    \normalfont
    \interlinepenalty\@M
    \Huge \bfseries  #1\par\nobreak
  }\vskip3cm}
  \makeatother
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