The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product.
The vectors $1$, $x$, and $x^2 - 1/3$ are orthogonal under the standard inner product for polynomials, $\int_{-1}^1 p(x)q(x)\ dx$.
While both collections meet their respective definitions of orthogonality, there seems to be some sense in which the former is more orthogonal than the latter.
No linear combination of $1$ and $x$ produces $x^2 - 1/3$, but the combination $-1/3$ of $1$ and $1$ of $x$, yielding $x - 1/3$, appears to make some progress toward the task. This isn't to claim that $x^2 - 1/3$ and $x - 1/3$ evaluate to similar values, but merely that if it were possible to reach $x^2 - 1/3$ by taking linear combinations of $1$ and $x$, this attempt would be a better start than, say, taking $-500$ of $1$ and $\pi$ of $x$. of By contrast, in the case of $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$, it feels like no progress can be made no matter what combination is chosen.
Is there some sense in which the former, $\mathbb{R}^3$ vectors are orthogonal to a stronger degree than the latter, polynomial vectors?