Inner Products, Length and Orthogonality

The dot product is the model inner product, written <u, v>. In R squared, (a, b) dot (c, d) = ac + bd. In R to the n, multiply matching components and add all n products. For nonzero u and v, cos theta = <u, v> / (length(u) length(v)). Vectors are orthogonal when their inner product is zero. The dot product of (3, -1) and (1, 3) is 3 - 3 = 0.

A real inner product must be symmetric: <u, v> = <v, u>. It must be linear in the first input: <a u + b w, v> = a<u, v> + b<w, v>. Symmetry gives linearity in the second input. Linearity in each input is called bilinearity. It must also be positive definite: <v, v> >= 0, with equality only for v = 0. These rules must hold for all vectors and real scalars. One successful numerical check cannot prove them.

An orthogonal set of nonzero vectors is independent. If c1 v1 + ... + ck vk = 0, take the inner product with vj. Orthogonality removes every term except cj<vj, vj>. Positive definiteness gives <vj, vj> > 0, so cj = 0. This works for every j. Thus a two-dimensional space can contain at most two mutually orthogonal nonzero vectors.

An orthonormal basis consists of mutually orthogonal unit vectors. The coordinate along ej is <v, ej>. For e1 = (3/5, 4/5), e2 = (-4/5, 3/5), and v = (2, 1), the coordinates are 2 and -1. Both basis vectors have length one and their dot product is zero. A negative coordinate means the component points opposite to that basis vector.

For an orthogonal basis that is not normalised, the coordinate along uj is <v, uj> / <uj, uj>. This follows by taking the inner product of v = sum cj uj with uj. For an orthonormal basis the denominator is one. Coordinates depend on direction: if e2 = (0, -1), then <(a, b), e2> = -b, not b.

Length in any real inner-product space is sqrt(<v, v>). For the dot product, length(a, b) = sqrt(a squared + b squared). Other inner products can give other lengths. The rule <u, v> = u1 v1 + 4 u2 v2 is symmetric and bilinear, and <v, v> = v1 squared + 4 v2 squared is positive for every nonzero v. In contrast, u1 v1 alone gives zero squared length to (0, 1), so it is not positive definite.