Coordinate systems

In order to identify a point in n-dimensional vector space it is necessary to define a basis $\beta$ for the space and an origin $\mathbf{O}$, which will be the point of space defined by the null vector:

$\beta = \left( \mathbf{e}_1, \, \mathbf{e}_2, \, \ldots , \, \mathbf{e}_n \right)$

$\mathbf{O} = \sum_{i=1}^n 0 \mathbf{e}_i$

Every point can now be identified with a vector or an n-tuple:

$\mathbf{P} = \sum_{i=1}^n \lambda_i \mathbf{e}_i$

$\mathbf{P} = \left( \lambda_1, \, \lambda_2, \, \ldots , \, \lambda_n \right)$

Some coordinate systems might have singularities where a point can be represented with different n-tuples, so a certain amount of care must be exercised when dealing with them.

See Also

• Cartesian Coordinates
• Cylindrical Coordinates
• Spherical Coordinates