Affine coordinate system

A rectilinear coordinate system in an affine space. An affine coordinate system on a plane is defined by an ordered pair of non-collinear vectors $ \mathbf e _ {1} $ and $ \mathbf e _ {2} $( an affine basis) and a point $ O $( the coordinate origin). The straight lines passing through the point $ O $ and parallel to the basis vectors are known as the coordinate axes. The vectors $ \mathbf e _ {1} $ and $ \mathbf e _ {2} $ define the positive direction on the coordinate axes. The axis parallel to the vector $ \mathbf e _ {1} $ is called the abscissa (axis), while that parallel to the vector $ \mathbf e _ {2} $ is called the ordinate (axis). The affine coordinates of a point $ M $ are given by an ordered pair of numbers $ (x, y) $ which are the coefficients of the decomposition of the vector $ \overline{OM}\; $ by the basis vectors:

$$ \overline{OM}\; = x \mathbf e _ {1} + y \mathbf e _ {2} . $$

The first number $ x $ is called the abscissa, while the second number $ y $ is called the ordinate of $ M $.

An affine coordinate system in three-dimensional space is defined as an ordered triplet of linearly-independent vectors $ \mathbf e _ {1} , \mathbf e _ {2} , \mathbf e _ {3} $ and a point $ O $. As in the case of the plane, one defines the coordinate axes — abscissa, ordinate and applicate — and the coordinates of a point. Planes passing through pairs of coordinate axes are known as coordinate planes.