# Why are vectors written like this: ||w||

• Femme_physics
In summary, || \cdot || is a symbol commonly used to represent the norm function on vectors. It is often written with single lines, which also denote the absolute value of a real number and the absolute value of a complex number. While it is sometimes used to denote a norm, it is generally considered bad form to do so. The symbol is meant to resemble absolute value because the magnitude of a vector is similar to the absolute value of a number.
Femme_physics
Gold Member
Is there a point to the double lines?

The symbol $|| \cdot ||$ (where the argument is placed where the dot is) is commonly used to represent the norm function on vectors.

There's no profound logic behind it then, right? Just the chosen symbol eh?

It's often written with single lines: |w|. Single lines also denote the absolute value of a real number, and the absolute value (also called the norm, magnitude or modulus) of a complex number. Maybe the notation started life there and spread to vectors.

Rasalhague said:
It's often written with single lines: |w|.

That symbol is used for absolute value and the determinant of a matrix. While it is sometimes used to denote a norm as well, but doing so is generally considered to be bad form.

Back to the OP, then: $||\text{whatever}||$ just a symbol with a (somewhat) well-agreed upon meaning. The typical meaning is the Euclidean norm. There are however many ways (an infinite number of ways) to define the length of a vector. All are norms.

First, vectors aren't written like that! The "magnitude" of a vector is written like that. And the symbol is intended to look like "absolute value" because the magnitude of a vector is similar to and is used like the absolute value of a number.

## 1. Why are vectors written with double bars?

The double bar notation for vectors, ||w||, is used to represent the magnitude of a vector, rather than its direction. This notation is common in mathematics and physics, where vectors are used to represent quantities such as force, velocity, and acceleration. By using double bars, it is clear that we are referring to the magnitude and not the direction of the vector.

## 2. Can't we just use single bars to represent magnitude?

Using single bars to represent magnitude can be confusing, as single bars are also used to denote the absolute value of a number. By using double bars for vectors, we avoid any confusion between the two concepts. Additionally, single bars are often used to denote the norm of a vector, which is a different mathematical concept than magnitude.

## 3. Why are there two bars instead of one?

The use of two bars is simply a convention in mathematics and physics. It is a visual representation of the concept of magnitude, and using two bars helps to differentiate it from other mathematical notations, such as single bars for absolute value or triple bars for norms.

## 4. Are there any other notations for vector magnitude?

Yes, there are other notations for representing vector magnitude. Some common alternatives include using the letter "v" with a single bar on top (v̅), or using the word "magnitude" with double bars around it (||magnitude||). However, the double bar notation (||w||) is the most widely recognized and accepted notation for representing vector magnitude.

## 5. Why do we need to represent vector magnitude separately from direction?

In many mathematical and physical applications, it is important to distinguish between the magnitude and direction of a vector. For example, in physics, we may be interested in the magnitude of a force acting on an object, rather than its direction. By representing magnitude separately, we can perform calculations and make predictions about the behavior of vectors without having to consider their direction.

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