Intuition for orthogonal vectors in $\Bbb R^n$ Two vectors in $\Bbb R^n$ are orthogonal iff their dot product is $0$. I'm aware that the dot product can be defined in other spaces, but to keep things simple let's restrict ourselves to $\Bbb R^n$.
Given that the idea of orthogonality is roughly to identify when two vectors have no "overlap", then apart from the fact that in $\Bbb R^2$ and $\Bbb R^3$ this corresponds to the geometrical notion of orthogonality, why is this chosen as the definition of orthogonality? Ideally give examples of concrete mathematical problems where this definition arises naturally.

Orthoganility intuitively could be reflcted as "no dependency". It is deeply anchored in the intuition of eigen-space and eigen-vector and eigen-value, eigen-... Recall that eigen means in German "the very geniune property of itself"  and this relates to a property which is absolutely genuine to $A$ and not shared with any $B$. So keeping also in mind that orthogonality of eigen-properties is relative. Indeed in $\Bbb R^n$ it ties depply with the dimmensions. There, dimmensions represent the number of possible eigen-properties (metaphoric: absolute selfish properties). The term orthogonality is lent indeed from the 90 degree or perpendicular $\perp$ visualisation of linear algebra but was by the mathematicians extended broadly to cover such type of relation that corresponds to eigen-properties under given constraints. From this point of view one should say orthogonal is a lent term that was used more and more expanded in its application to genuine independency and not restricted to be set identical to the term perpendicular.
Example: generally applicable you can test any $A$ and $B$ relatively to each other and under your given constraints with respect to orthogonality. A beautiful one might be that two theorems $T_1$ and $T_2$ could be orthogonal.
