## Quick Reference

Points in the plane can be specified by means of Cartesian coordinates (*x, y*), and points in 3-dimensional space may be assigned Cartesian coordinates (*x, y, z*). It can be useful, similarly, to consider space of *n* dimensions, for general values of *n*, by defining a point to be given by *n* coordinates. Many familiar ideas from geometry of 2 and 3 dimensions can be generalized to space of higher dimensions. For example, if the point *P* has coordinates (*x*_{1}, *x*_{2},…, *x*_{n}), and the point *Q* has coordinates (*y*_{1}, *y*_{2},…, *y*_{n}), then the distance *PQ* can be defined to be equal to

Straight lines can be defined, and so can the notion of angle between them. The set of points whose coordinates satisfy a linear equation *a*_{1}*x*_{1}+*a*_{2}*x*_{2}+…+*a*_{n}*x*_{n}=*b* is called a hyperplane, which divides the space into two half-spaces, as a plane does in 3 dimensions. There are other so-called subspaces of different dimensions, a straight line being a subspace of dimension 1 and a hyperplane being a subspace of dimension *n*−1. Other curves and surfaces can also be considered. In *n* dimensions, the generalization of a square in 2 dimensions and a cube in 3 dimensions is a hypercube.

*Subjects:*
Mathematics.

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