# How to understand the two vertical lines ||

• Jun 30th 2010, 02:00 AM
dervast
How to understand the two vertical lines ||
Hello I read the following equation
Imageshack - log.gif
but I do not understand what the
|| (what ever is inside) ||2 (with the 2 as a small number -- called index?)

Best Regards
Alex
• Jun 30th 2010, 02:07 AM
Zaph
That would be the 'length'
• Jun 30th 2010, 02:51 AM
drumist
$\displaystyle || \mathbf{a} ||$ is a notation that represents the "length" of a vector $\displaystyle \mathbf{a}$. In your expression, $\displaystyle \mathbf{x} - \mathbf{x}_r$ is the vector whose length is being calculated.

Since "length" can be defined in a number of ways, the subscript $\displaystyle 2$ indicates that the length is calculated using the Euclidean norm. So, $\displaystyle ||\mathbf{x} - \mathbf{x}_r||_2$ means that you should find the length of the vector $\displaystyle \mathbf{x} - \mathbf{x}_r$ as defined by the Euclidean norm.

The Euclidean norm is the most intuitive definition of length. For example, if you have a vector $\displaystyle (4,-3)$, its Euclidean length would be $\displaystyle 5$. You can conceptualize this as the distance between the point $\displaystyle (4,-3)$ and the origin.
• Jun 30th 2010, 04:20 AM
dervast
Quote:

Originally Posted by drumist
$\displaystyle || \mathbf{a} ||$ is a notation that represents the "length" of a vector $\displaystyle \mathbf{a}$. In your expression, $\displaystyle \mathbf{x} - \mathbf{x}_r$ is the vector whose length is being calculated.

Since "length" can be defined in a number of ways, the subscript $\displaystyle 2$ indicates that the length is calculated using the Euclidean norm. So, $\displaystyle ||\mathbf{x} - \mathbf{x}_r||_2$ means that you should find the length of the vector $\displaystyle \mathbf{x} - \mathbf{x}_r$ as defined by the Euclidean norm.

The Euclidean norm is the most intuitive definition of length. For example, if you have a vector $\displaystyle (4,-3)$, its Euclidean length would be $\displaystyle 5$. You can conceptualize this as the distance between the point $\displaystyle (4,-3)$ and the origin.