When I first got into MA101 in college, my first calculus course, I was confounded when the professor referred to equations involving a multitude of dimensions each of which was orthogonal to all of the others. Taking the word "orthogonal" to be a synonym for the word "perpendicular", I couldn't envision the idea of more than three dimensions other than in what I thought must be some abstruse, arcane or even imaginary sense and that really bothered me.

I never claimed to be a great math student and I guess this personal anecdote proves the point. What I failed to grasp was the following.

Imagine that I have the one particular "point". The physical position can be described in terms of the x-coordinate, the y-coordinate and the z-coordinate which gives us three dimensions. So far, so good.

At that same exact point however, we may have some particular temperature as a fourth dimension, some particular barometric pressure as a fifth dimension (not the musical group), some particular time as a sixth dimension, some particular level of illumination as a seventh dimension and maybe more if we keep thinking about this.

The key point about these seven dimensions is that a change in value of any one of them does not require that there be a resultant change of value of any of the others and this is orthogonality. Each of these seven dimensions is orthogonal to the other six.

I finally understood that perpendicularity refers to a three-dimensional subset of orthogonality but it took me a long while and it was a real conceptual impediment for me in that MA101 class.