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Let $V$ be a vector space over $\mathbb R$. Given $u, v\in V$ where $u\neq 0$, there is a unique vector $w\in V$ such that:

I) $w$ is parallel to $u$;

II) $w-v$ is perpendicular to $u$.

In fact, if we let

$$w=\frac{\langle u, v\rangle}{\langle u, u\rangle}u$$

then $w$ is parallel to $u$ by the very definition, and

$$\langle u, w-v\rangle=\langle u, w\rangle-\langle u, v\rangle =\frac{\langle u, v\rangle}{\langle u, u\rangle} \langle u, u\rangle-\langle u, v\rangle=\langle u, v\rangle=0.$$

As to the uniqueness, if $w^\prime$ has the same properties as $w$, then

$$w^\prime = \alpha u$$

for a given scalar $\alpha\in\mathbb R$. Then,

$$0=\langle u, w-v\rangle=\langle u, w\rangle-\langle u, v\rangle=\alpha \langle u, u\rangle-\langle u, v\rangle$$

what implies

$$\alpha=\frac{\langle u, v\rangle}{\langle u, u\rangle}u$$ so that

$$w^\prime=\frac{\langle u, v\rangle}{\langle u, u\rangle}u=w.$$

The vector

$$\textrm{pr}_{u}(v)=\frac{\langle u, v\rangle}{\langle u, u\rangle}u$$

is called the orthogonal projection of $v$ over $u$.

What is the relevance of $\textrm{pr}_u(v)$ after all?

I know, for instance, that if $W\subset V$ is a subspace and $\{w_1, \ldots, w_m\}$ is a basis for $W$, then the vector

$$\textrm{pr}_W(v)=\frac{\langle w_1, v\rangle}{\langle w_1, w_1\rangle}w_1+\ldots + \frac{\langle w_n, v\rangle}{\langle w_m, w_m\rangle}w_m$$

provides the vector on $F$ which is closest to $v$. But again, and so what?

Can anyone point me out some further implications of the existence of an orthogonal projection? Up to this point, I see it as a tool, but it is not clear what is that good for.

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    $\begingroup$ Least Squares Regression is just orthogonal projection on the duel space. It also allows you to approximate functions by projecting onto some class of orthogonal functions. Orthogonal projections are also used extensively in computer graphics and drafting. In coding theory certain projections will be able to detect and correct errors in a transmission. $\endgroup$ Dec 20, 2021 at 20:28

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"... 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.

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