A vector as a directed line segment in a chosen coordinate system, by definition, should have the same number of components as there are coordinate axes. In a Cartesian system, a vector has three orthogonal components which are also called real components. Altogether these three components define one among eight properties for a principle of directional invariance. In basis notations, these eight directional properties are (1, 1, 1), (-1, 1, 1),(1, -1, 1), (1, 1, -1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1), (-1, -1, -1). The reality of these components is based on the possibility of orthogonality. By adding another axis, say the time axis, it is physically impossible to have all four axes orthogonal to each other. Therefore, by definite rules of mathematical logic, the 4th axis must be imaginary as well as its component. When an axis is in imaginary space, it can be made orthogonal to any real axis as long as it remains in imaginary space. The moment the imaginary axis crosses the real axis its metric property vanishes. The vanishing point becomes the coordinate origin of a complex plane http://en.wikipedia.org/wiki/Complex_plane called the Argand diagram http://mathworld.wolfram.com/ArgandDiagram.html. However, the origin (0, 0, 0) is not one of the directional properties and could not be used to represent them. The deficiency of any multi-dimensional coordinate system (including tensor representations) to represent the eight directional properties for the principle directional invariance initiated the search for alternative representations. The best representation is found by the use of Hadamard matrices.