After a quick review of how vectors are represented in the plane, let’s get to know the concept of the spatial coordinate system and the possibilities of decomposing vectors in space.

In the plane, the position of points is given using the Cartesian rectangular coordinate system, which is based on two mutually perpendicular basis vectors. Any planar vector can be represented as a linear combination of these two basis vectors. This representation is unique. 

The vector a uniquely determines the ordered pair of numbers (α₁;α₂). 
Conversely, the ordered pair (α₁;α₂) uniquely determines the vector a.

For spatial vectors, two basis vectors are not sufficient; a third one is needed. This third basis vector is denoted by k. This unit vector is perpendicular to both i and j.
However, in space, there are two possible vectors that satisfy the perpendicularity condition. The vector k is the one for which, when viewed from its endpoint, i must be rotated counterclockwise through j to obtain k. The line passing through the origin and having the same direction as k is called the z-axis. The vectors i, j, and k (in this order) form a right-handed system.

In space, we therefore need a rectangular solid whose one vertex is the origin O, its edges are parallel to the coordinate axes, and whose diagonal starting from O is exactly a.
This vector a can be written in the following form:

a = α₁i + α₂j + α₃k

Each unique linear combination defines a distinct rectangular solid – and therefore a distinct diagonal vector. Thus, every spatial vector a uniquely determines an ordered triple of numbers (α₁;α₂;α₃), and every ordered triple (α₁;α₂;α₃) uniquely determines a spatial vector a.