Video which introduces the inner product as an operation that allows us to solve the problem of "finding coefficients" in finite-dimensional vector spaces in a simple and scalable manner through the use of orthogonal bases, and then uses this result to give a geometric interpretation of the inner product.
Video which introduces the notion of norm and its relationship with inner products, as well as that of orthonormal bases, which simplify the solution to the problem of "finding coefficients" even further. (Currently in production in collaboration with Animathica; the video script and animation code can be found in this repository.)
Video which illustrates the orthogonalization and orthonormalization processes in finite dimensional inner product spaces and explains the Gram-Schmidt Theorem.