Vertical dots are a common way of indicating a range of elements in a list or a sequence. They can be used to signify incomplete or omitted parts of a list, or they can be used to show the progression of a series. In LaTeX, there are several ways to generate vertical dots, the most common of which involve the use of the \vdots and \cdots commands. Here, we will discuss the different ways to use vertical dots in LaTeX with code examples.
Vertical Dots Using \vdots
The \vdots command is used to generate vertical dots in LaTeX. It is typically used to show an incomplete or omitted part of a sequence or a list. Here is an example of how to use \vdots in LaTeX:
\begin{align*}
a_1 &+ a_2 + a_3 + \cdots + a_n \
&\vdots \
a_{n2} &+ a_{n1} + a_n \
\end{align*}
In this example, the \vdots command is used to indicate that there are more terms in the sequence that are omitted for brevity.
Vertical Dots Using \cdots
The \cdots command is used to generate horizontal dots in LaTeX. It is typically used to show a continuous or repetitive pattern in a sequence or a list. However, with some special formatting, it can also be used to generate vertical dots. Here is an example of how to use \cdots to generate vertical dots in LaTeX:
\begin{align*}
a_1 &+ a_2 + a_3 + \cdots + a_n \
&\raisebox{1ex}{$\cdots$} \
a_{n2} &+ a_{n1} + a_n \
\end{align*}
In this example, the \cdots command is used to generate a horizontal dot that is then placed vertically using the \raisebox command. This will produce a similar effect as \vdots, but with a different formatting for the dots.
Vertical Dots Using amstext
The amstext package provides an alternative method for generating vertical dots in LaTeX. This package provides the \dotsb command, which can be used to generate vertical dots in a list or a sequence. Here is an example of how to use \dotsb in LaTeX:
\begin{align*}
a_1 &+ a_2 + a_3 + \dotsb + a_n \
&\dotsb \
a_{n2} &+ a_{n1} + a_n \
\end{align*}
In this example, the \dotsb command is used to generate vertical dots that are similar in appearance to \vdots. However, this command provides more flexibility in terms of customization and formatting.
Conclusion
In summary, there are several ways to generate vertical dots in LaTeX. The most common methods involve using the \vdots and \cdots commands, as well as the amstext package's \dotsb command. Each of these methods provides a different way to indicate the presence or absence of elements in a sequence or a list. By choosing the appropriate method and formatting for vertical dots, you can improve the readability and clarity of your LaTeX document.
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Vertical Dots Using \vdots and \cdots:
The vertical dots using \vdots are commonly used to indicate omitted rows of a matrix or missing terms in a sequence, while the horizontal dots using \cdots are usually used to indicate the repetition of terms or pattern in a sequence. The difference between the two commands is that \vdots command generates vertical dots in a column while the \cdots command generates horizontal dots in a row, but with specific formatting, \cdots can be used to generate vertical dots.
In some cases, you may need to use both vertical and horizontal dots to represent a pattern in a sequence or a matrix. For example, consider the following matrix:
\begin{bmatrix}
1 & 2 & 3 & \cdots & n \
4 & 5 & 6 & \cdots & n+3 \
7 & 8 & \vdots & & \vdots \
\vdots & \vdots & & & \vdots \
3n2 & 3n1 & \cdots & \cdots & 3n
\end{bmatrix}
Here, we have used the \vdots and \cdots commands to generate the vertical and horizontal dots, respectively. This matrix represents a pattern where each row has a constant addition to the previous row.
Vertical Dots Using amstext:
The amstext package provides a powerful way of creating vertical dots, called \dotsb, that allows you to customize the appearance of vertical dots. This package also provides other commands like \dotsc, \dotsi, and \dotsm for creating different kinds of dots. The \dotsb command is used in mathematical texts to create dots that separate summation terms or products.
For example, if you have a summation or product expression with multiple terms, you can use \dotsb to indicate the omitted terms, as shown in the example below:
$$\sum_{i=1}^{n} f(x_i) = f(x_1) + f(x_2) + \dotsb + f(x_n)$$
In this example, we have used the \dotsb command to indicate that there are more terms in the series that are omitted for brevity. The dots are placed in the vertical direction to indicate that the terms are on the same line.
Conclusion:
In conclusion, vertical dots are an essential tool while typesetting mathematical expressions. It helps readers quickly understand the presence or absence of terms in a sequence or matrix. In LaTeX, there are various ways to generate vertical dots. The most common and straightforward method is using the \vdots command for vertical dots and the \cdots command for horizontal dots. Besides, the amstext package provides more customization options for creating vertical dots. By choosing the appropriate method and formatting, you can improve the readability and clarity of your LaTeX document.
Popular questions
Sure, here are five questions with answers about "Vertical Dots in LaTeX with Code Examples":

What is the primary purpose of using the \vdots command in LaTeX?
Answer: The \vdots command is primarily used to generate vertical dots in a column, indicating omitted rows of a matrix or missing terms in a sequence. 
How can you use the \raisebox command with the \cdots command to generate vertical dots in LaTeX?
Answer: The \cdots command generates horizontal dots in a row, so you can place them vertically using the \raisebox command. For example, you could use \raisebox{1ex}{$\cdots$} to create vertical dots that are aligned with the centerline of the previous row. 
What is the purpose of the amstext package in LaTeX?
Answer: The amstext package provides additional mathematical symbols and features, including the \dotsb command, which can be used to generate customized vertical dots in a list or a sequence. 
Can you use both \vdots and \cdots commands in the same matrix to represent a pattern?
Answer: Yes, you can use both \vdots and \cdots commands in the same matrix to represent a pattern. For example, you could use \vdots to represent missing rows and \cdots to represent a pattern between the existing rows. 
Why are vertical dots important in LaTeX?
Answer: Vertical dots are important in LaTeX because they help to visually indicate the presence or absence of terms in a matrix or a sequence, which can greatly improve the readability and clarity of mathematical expressions.
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