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

[Femme_physics]

Is there a point to the double lines?

[Hurkyl]

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

[Femme_physics]

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

[Rasalhague]

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.

[D H]

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.

[HallsofIvy]

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.