Dot Operator

The Mathematical Symbol "Dot Operator (⋅)"

The Dot Operator (⋅): An Insight

The Dot Operator, symbolized as ⋅ is a mathematical notation frequently employed in various mathematical contexts. One of its primary usages is in the realm of vector calculus, where it represents the dot product. Additionally, it can denote multiplication in some scenarios, especially when traditional multiplication signs might be confusing. In this article, we'll dive deep into the significant applications of this symbol with a pair of examples for each use.

1. Vector Calculus: Dot Product

In vector calculus, the ⋅ notation denotes the dot product (or scalar product) of two vectors. This operation yields a scalar value and is defined for two vectors as the product of their magnitudes and the cosine of the angle between them.

Example 1:

Given two vectors \(\mathbf{A}\) and \(\mathbf{B}\), their dot product is represented as:

\(\mathbf{A} ⋅ \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta)\)

Where \( \theta \) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\).

Example 2:

For vectors \(\mathbf{A} = [a_1, a_2]\) and \(\mathbf{B} = [b_1, b_2]\) in a 2-dimensional space:

\(\mathbf{A} ⋅ \mathbf{B} = a_1 \times b_1 + a_2 \times b_2\)

2. Multiplication Representation

Occasionally, the ⋅ symbol is used to represent multiplication, especially in contexts where the traditional multiplication signs (like \( \times \)) might cause confusion.

Example 1:

When dealing with algebraic variables:

\(a ⋅ b\), which simply means \(a \times b\).

Example 2:

In scalar multiplication:

\(5 ⋅ 3 = 15\)

These instances showcase the versatility of the Dot Operator in various mathematical scenarios. As with many mathematical symbols, context is crucial for interpreting its precise meaning.

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Codes for the ⋅ Symbol

The Symbol	⋅	copy
Alt Code	Alt 8901	copy
HTML Code	⋅	copy
HTML Entity	⋅	copy
CSS Code	\22C5	copy
Hex Code	⋅	copy
Unicode	U+22C5	copy

How To Insert the ⋅ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ⋅ symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8901. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML Text	Output
<b>My symbol: &#8901;</b>	My symbol: ⋅

(Method 4) Use the HTML Entity Code (for webpages).

HTML Text	Output
<b>My symbol: &sdot;</b>	My symbol: ⋅

(Method 5) Use the CSS Code (for webpages).

CSS and HTML Text	Output
<style> span:after { content: "\22C5";} </style> <span>My symbol:</span>	My symbol: ⋅

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML Text	Output
<b>My symbol: &#x22C5;</b>	My symbol: ⋅

On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22C5 to place the ⋅ symbol on your canvas. For example:

JavaScript Text
const x = "0x"+"E9" ctx.fillText(String.fromCodePoint(x), 5, 5);
Output
⋅

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ⋅ is U+22C5. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:

Type	Output
22C5 [Hold down Alt] [Press x]	⋅ (The 22C5 turns into ⋅. Note that you can omit any leading zeros.)

In JavaScript, the syntax is \uXXXX. So, our example would be \u22C5. (Note that the format is 4 hexadecimal characters.)

JavaScript Text	Output
let str = "\u22C5" document.write("My symbol: " + str)	My symbol: ⋅