{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

A rather spontaneous revelation occurred to me while lying on the floor on my belly involved in my daily ritual of enduring existence, that is, working through math(s) exercises.

I found a particular set of problems extremely boring, to the point where I felt like a child in grade school going through the motions with very little brain power exerted.

I am in the process of putting together this solution manual, so I am not skipping any problems.

Well, I found that there was an alternative approach, and using this method made the boredom-inducing problems a little more interesting.   Is this not in some sense a defamiliarization process, where something familiar to the point of boredom can be made more interesting using a more exotic or more elegant method of calculation?

The problems were trivial, where the student is asked to derive a scalar equation after having found the slope from the given points P and S.

Now, most students, including myself, would use the familiar point-slope form of an equation:

y - y1 = m(x - x1), where (x1, y1) are the coordinates of either P or S.
One then manipulates this algebraically into the scalar form a*x + b*y = c (with no fractions as coefficients).

I was so bored that I wondered why I was spending my life this way.  My brain had turned off, and I was just going through the motions.

And yet, the purpose of this particular drill was not just to derive the scalar equation by any means necessary, but, I suppose, to use the algebra of vectors and a theorem such as X dot n = P dot n, where X = (x,y), n = (a,b) is the vector normal (perpendicular) to the line, and "dot" means taking the "inner product" (dot product).

What made these all-too-boring and routine exercises so much more interesting was what I want to refer to as "mathematical defamiliarization".

Rather than using the familiar and boring method, using the unfamiliar method seems more elegant and sophisticated.  It had more MEANING.

Given points P and S, you can find the slope m, true; but you can also find the normal vector n.

n will be perpendicular to (P-S).

Once you have n = (a,b), then you derive the scalar equation with:  (x,y) dot (a,b) = (x1, y1) dot (a,b)

a*x + b*y = a*x1 + b*y1

This actually felt more elegant than simply using the point-slope form: y - y1 = m(x - x1).

I think that this quite accidentally serves as another example of the benefits of defamiliarization in mathematics.  It can make things less boring, less routine, and more interesting, which makes it more satisfying.