# Definition:Taxicab Metric/General Definition

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## Definition

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces.

Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.

The taxicab metric on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as:

$\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n d_{i'} \left({x_i, y_i}\right)$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.

### Real Vector Space

This metric is often seen in the context of the real vector space $\R^n$:

The taxicab metric on $\R^n$ is defined as:

$\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n \left\vert {x_i - y_i}\right\vert$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.