koordinaatioyksikkö

Koordinaatioyksikkö is a Finnish term referring to a unit vector aligned with a coordinate axis in a coordinate system. It represents a direction along one of the axes and has length one. Koordinaatioyksiköt are the building blocks for expressing general vectors as linear combinations of axis-aligned directions.

In a Cartesian 2D or 3D system, the standard koordinaatioyksiköt are denoted e_x, e_y (and e_z in 3D). A typical representation is v = v_x e_x + v_y e_y (+ v_z e_z in three dimensions), where v_x, v_y, and v_z are the vector’s components along the respective axes. The koordinaatioyksiköt form an orthonormal basis: e_i · e_j = δ_ij, meaning they are mutually perpendicular and each has unit length.

Koordinaatioyksikköiden käsite is central in linear algebra, vector calculus and physics, enabling decomposition of vectors, projections, and coordinate transformations. They are also referred to as the standard basis or canonical basis in many contexts. While the term typically denotes axis-aligned unit vectors, it is distinct from unit vectors in arbitrary directions (for example, a unit vector in the direction of a given vector v, which is v/|v|).

Examples include e_x = (1, 0, 0), e_y = (0, 1, 0), and e_z = (0, 0, 1) in Cartesian coordinates. In other coordinate systems, corresponding koordinaatioyksiköt exist for the directions along the system’s axes, such as radial or angular directions in cylindrical or spherical coordinates. See also standard basis, unit vector, and coordinate systems.