Orthogonality - John Dunn, Consultant, Ambertec, P.E., P.C.

At the following URL, we find several definitions of the word "orthogonal", but the one I want to call attention to is the fifth definition, "statistically independent".  

Please see:  http://www.merriam-webster.com/dictionary/orthogonal

Back in my college days, this particular definition  would have helped me resolve a personal mental- block about mathematical analyses that involve more than three dimensions. How could there possibly be more than three dimensions? 

The preconception that was getting in my way was that we can think of things as having no more than three physical dimensions, length, width and height.  Very simple, right?

However, where the variables in some presented analyses were said to be "orthogonal",  I had this unfortunate student tendency to think of that word "orthogonal"  as a synonym for "perpendicular".  Since all we have are length, width and height, so my thinking went, how can we possibly address any analysis involving more than three dimensions?

I finally broke this mental logjam by realizing that "perpendicular" is just a subset of "orthogonal" in the following sense.

Let's say that I have an object located at some point in space. That position is defined by three spatial coordinates, the x-axis value, the y-axis value and the z-axis value. A value change in any one of these three coordinate values does not require that there be a change of value in the other two. These three things are statistically independent as in the above definition. Those values are descriptors of perpendicular items and of orthogonal items.

However, at that hypothetical point in space, we can also note that there is time (a fourth dimension), temperature (a fifth dimension), barometric pressure (a sixth dimension) and maybe a few more dimensions that have not, for the moment, come to mind.

Each of these six items is statistically independent of the other five. Therefore, these six items are mathematically orthogonal to each other. Never mind the word "perpendicular".

Do you remember what it feels like when an epiphany of comprehension suddenly sets in?