Although mathematically speaking there is no limit to the number of dimensions that can be defined using the Cartesian orthogonal system of coordinates, perpendicularity must always be done in one dimension. One-dimensional entities are lines whether straight or curved. Two given straight lines can only intersect at a point of zero dimensions. However, the shortest line must be defined as composed of two zero dimensional points separated by zero distance which are also called the end points of the shortest line as previously defined.

Two equal shortest lines AB and BC intersect at the point B are perpendicular if and only if AB=BC=1 as the unit length and such that the line AC is equal to the square root of 2. Given three unit shortest lines AB, BC, and BD intersecting at B, these are mutually perpendicular if and only if the lines AC, AD, and CD all equal to the square root of 2. Given four unit shortest lines AB, BC, BD, BE intersecting at B, again these are mutually perpendicular iff the lines AC, AD, AE, CD, CE, and DE are all equal to the square root of 2. Clearly, it can be noted that the number of lines equal to square root of 2 becomes twice as much for 4 dimensions as it is needed for 3 dimensions. For five equal shortest lines AB, BC, BD, BE, and BF intersecting at B, perpendicularity requires that AC, AD, AE, AF, CD, CE, CF, DE, DF, and EF must all equal to the square root of 2. The number of linear conditions for dimensional perpendicularity 1, 3, 6, and 10 is equivalent to the third diagonal numbers of Pascal’s triangle. Base on this diagonal, the number of linear conditions of perpendicularity can be predicted to any given spacetime dimension.