AoPSWiki
Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

Euclidean metric

From AoPSWiki

The Euclidean metric on \mathbb{R}^n is the standard metric on this space. The distance d(\mathbf{x, y}) between two elements \mathbf{x} = (x_1, x_2, \ldots, x_n) and \mathbf{y} = (y_1, y_2, \ldots, y_n) is given by d(\mathbf{x, y}) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \ldots + (x_n - y_n)^2}. It is straight-forward to show that this is symmetric, non-negative, and 0 if and only if \mathbf{x = y}. Showing that the triangle inequality holds true is somewhat more difficult, although it should be intuitively clear because it is properties of the Euclidean metric which motivate the definition of a metric.

Proof of the triangle inequality

See Also

This article is a stub. Help us out by expanding it.

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Euclidean_metric"
Add a glimpse of the Art of Problem Solving Forum to your own site!
Click here for details!
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us