By Chandrashekar (Chandra) Tamirisa, (On Twitter) @c_tamirisa
(The idea for this technical paper came about when I was tutoring my nephew Vasanth Neppalli for the Scholastic Aptitude Test, SAT)
The axiomatic Euclidean planar space is defined as two axes, the horizontal by convention, X, and the vertical by convention, Y, at right angles or orthogonal to each other.
The slope of any straight line in that 2-dimensional coordinate space is defined as the ratio of the positive or negative difference of the Y coordinates over the positive or negative difference of the X coordinates. If the ratio is positive the straight line is upward sloping and if it is negative the straight line is negative sloping.
Proofs, also by convention, exist that, in this space, the product of the slopes of any two perpendicular or orthogonal straight lines – a straight line defined as a line connecting two points in space (an arc requiring three points) – is -1.
The statement of this paper is to point out that this proof does not hold for the axiomatic 2-dimensional Euclidean planar space defined by the orthogonal infinitely extending straight lines X-Y: the product of the slopes of X and Y axes in any direction is not -1.
This paper, therefore, further contends that the product of the slopes of the system of any two orthogonal straight lines displaced at an angle to X-Y space is also not -1.
In conclusion, the product of the slopes of any two orthogonal straight lines in Euclidean planar space is not -1.
QED
