Given the inner product of two polynomials $p(X), q(X) \in P(d)$, where $P(d)$ is the vector space of all polynomials of degree less than or equal to d, with real coefficients, and using the inner product $$\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$$
How can the angle $$\cos(\alpha)=\frac{\langle p(x),q(X) \rangle}{\|p(X)\|\|q(X)\|}$$
be interpreted geometrically?
The angle, the normalized inner-product, can be seen as a measure on the scale from 0 to 1 of how "orthogonal" (independent) or "parallel" (dependent) are the two polynomials.