When should we use a blank line?

Question

I only know that we have to make at least one blank line between two adjacent paragraphs.

My question is:

What is the proper way to make use of a blank line ?

For example:

1. Is it recommended to make a blank line between heading command call, e.g. \chapter{}, and the first text content as follows?

   \chapter{Introduction}
   
   In this chapter we ....
   

   or

   ... that is proven true.
   
   \chapter{Limit}
   

2. Is it recommended to make a blank line between two consecutive display equations as follows?

   \[ \sin x \] 
   
   \[ \cos x \]
   

3. Is it recommended to make a blank line between a sentence and a display equation as follows?

   Assume we have
   
   \[
     \cos x
   \]
   

   or

   \[
     \cos x
   \]
   
   that satisfies ...
   

4. etc...

---

Edit 1:

1. How about \bigskip, \vspace{}? Does each of them needs one blank line before and one blank line after as follows:

   Yes. This is a book.
   
   \bigskip
   
   Commander, please open the silo!
   

   or

   Yes. This is a book.
   
   \vspace{1km}
   
   Commander, please open the silo!
   

Answer 1

I can tell you what I'd do in these 3 cases:

1.: After \chapter{} and similar command I'd always use a blank line as it makes the document structure clearer. Moreover, "In this chapter ..." is the start of a paragraph.

2.: I wouldn't use a blank line here if both displays belong to the same paragraph. However, I usually wouldn't use two consecutive displays at all; I'd use align* or gather* instead.

3.: In that particular case don't use blank lines as everything is within one paragraph. Speaking of clarity of the document structure, I'd say that

Assume we have    
\[
  \cos x
\]
that satisfies ...

is clear enough. Only use a blank line after the display if the paragraph ends after the display.

4.: As for your extra question about vertical spacing commands: In the context you provide, I'd always surround them with blank lines for clarity of the document structure. I sometimes use them before a display like this; then you mustn't put a blank line:

... some text
\vspace{-1ex}
\[
  ...
\]

I should point out that manual corrections like in this last example are rarely needed, and you should only use them if you're sure that you know what you're doing.

Answer 2

it gives wrong vertical spacing, when you use blank lines before, between and after display math environments. Compare the following output, where the equation environment has not the same spacing over and under the math expression.

\documentclass{article}
\begin{document}

\begin{minipage}{0.49\linewidth}
The line above the equation.

\[ f(x)=\prod_{i=1}^n\left(i-\frac{1}{2i}\right) \]

The line below the equation.
\end{minipage}\hfill\begin{minipage}{0.49\linewidth}
The line above the equation.

\begin{equation} 
f(x)=\prod_{i=1}^{n}\left(i-\frac{1}{2i}\right) 
\end{equation}

The line below the equation.
\end{minipage}

\bigskip
\begin{minipage}{0.49\linewidth}
The line above the equation.
%
\[ f(x)=\prod_{i=1}^n\left(i-\frac{1}{2i}\right) \]
%
The line below the equation.
\end{minipage}\hfill\begin{minipage}{0.49\linewidth}
The line above the equation.
%
\begin{equation} 
f(x)=\prod_{i=1}^{n}\left(i-\frac{1}{2i}\right) 
\end{equation}
%
The line below the equation.
\end{minipage}
\end{document}

Answer 3

For your new part 4, the answer is it doesn't matter. \bigskip expands to \vspace\bigskipamount and \vspace contains code that deals with it be used in horizontal mode (namely, it inserts the \vskip into the surrounding vertical list after the current line using \vadjust).