Philosophical question on visualizing basis vectors as orthogonal

Question

In the space $\mathbb{R}^n$, by definition, the component representation of $e_1, e_2, ..., e_n$ with respect to basis $\{ e_1, e_2, ..., e_n \}$ is (1, 0, ..., 0), (0, 1, 0, ..., 0), ..., (0, 0, ..., 1).

Without equipping this space with a choice of inner product, it is not defined whether any pair of these basis vectors are orthogonal, nor is their length defined.

Yet it is a common habit to visualize the $\mathbb{R}^3$ vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) along a left-right horizontal axis, a front-back horizontal axis, and a vertical axis, and furthermore imagine the tick at '1' to be the same distance from the origin along all of these axes.

Is the act of visualizing these this way somehow inducing the choice of the dot product as the inner product? Or, is there any way to rationalize and connect the visual imagination being performed here with the mathematical assumptions it induces?

Answer 1

Is the act of visualizing these this way somehow inducing the choice of the dot product as the inner product?

Yes. The general inner product of the space $\mathbb{R}^n$ (considered to be column vectors) is $$ <x, y>= x^Tgy $$ where $g$ is some symmetric matrix. For example, in $\mathbb{R}^2$ $$ <x, y>= a x_1 y_1 + b x_2 y_2 + c (x_1 y_2 + x_2 y_1). $$

Thus the basis vectors $(1,0)$ and $(0,1)$ may not be unitary or orthogonal. The matrix $g$ can be considered to contain, as elements, the dot products of vectors, and is known in differential geometry as the metric tensor. To be a true inner product we require $g$ to be positive definite (so that the magnitude of all non-zero vectors is positive), but in differential geometry this requirement is often dropped.

Answer 2

As you noted, the vectors of any basis in $\mathbb{R}^3$, represented in the same ''themself'' basis, have components $(1,0,0)^T$, $(0,1,0)^T$ and $(0,0,1)^T$.

More, for any basis ve can define an inner product in $\mathbb{R}^3$ such that the vector of the basis are orthogonal ( see here).

Usually we represents the vectors $(1,0,0)^T$, $(0,1,0)^T$ and $(0,0,1)^T$ as a triple of orthogonal vector like the three sides of a cube, but this is only a consequence of a bias deriving from our physical experience, because in our world there is a ''vertical'' direction and a ''horizontal'' plane (the horizon) determined by gravity.

I don't see a pure mathematical reason to represents ''orthogonality'' only in this usual way..... But maybe I'm wrong. I have a question about this but without an answer.