On the relevance of $\textrm{pr}_u(v)$?

Solution 1:

"... provides the vector on $W$ which is closest to $v$... so what?"

Finding the closest approximation to something is a very important problem to be able to solve. For example, in statistics I might wish to model a data set using a linear model, say linear regression. It is not easy to determine how I might find a "line of best fit", but if I can precisely compute the vector in the span of my modelling parameters which is closest to the "vectors" (data points with potentially multiple coordinates) of my data set, I can go home happy. I could also wish to find an optimal function in a function space, given some restrictions, if those restrictions formed a vector subspace.

In maths generally, in mechanics specifically, it is useful to know how much of a quantity "lies" in a given direction. I may wish to split the force acting on a body into three orthogonal components, based on a local coordinate system, to solve a mechanics problem, such as the magnitude of the friction in one direction, or perhaps the tension on a cord attached to the body in another direction, etc.

Orthogonal bases are nice in linear algebra. Being able to express a vector in terms of them is useful, knowing it is possible is useful in proofs - I have seen proof authors say "w.l.o.g express $v$ in terms of an orthogonal basis" and this will simplify the proof. To reiterate what I believe is the main point, in consideration of a subspace of interest I will often want to solve optimisation problems which are elegantly (and computationally efficiently) solved with orthogonal projection. Orthogonal projectors, as operators, are also nice since they satisfy self adjointness.