Dot product

In analytic plane geometry, the basic tasks are measuring lenghts and angles; everything else depends on them. The basic tools for these tasks is the dot product. The standard Euclidean inner product is

$$\mathbf{x}\cdot \mathbf{y}=\sum _{i=1}^n{x}_i{y}_i=x_1{y}_1+\dots +x_n{y}_n\text{\hspace{1em}1.5.2}$$

This can be written very succinctly using Einstein summation notation as

From the inner product we get the lenght, or Euclidean norm, of a vector,

$$|\mathbf{x}|=\sqrt{\mathbf{x}\cdot \mathbf{x}}$$

The lenght of a vector x is the square root of the dot product of that vector with itself.

The angle θ between two vectors x and y may be determined from

In order to understand why the algebraic dot product between two vector yields the trigonometric cosine value, one can use the cosine law from Euclidean geometry. Consider the triangle created by the origin, x and y. We want to find the angle θ between x and y. The Euclidean side lenghts of this triangle are a = |x|, b = |y|, and c = |x − y|. The cosine law provides a formula for the angle θ in terms of side lenghts as follows

where θ is the angle between the vectors and |x| is the norm.

Dot product on ℂⁿ

The dot product on ℂⁿ is defined by

Properties of the dot product

The following proposition shows that the dot product satisfies the properties of linearity, symmetry, and positivity.

Proposition 1.8.1 The dot product on ℝⁿ satisfies the following properties for all v, w, u ∈ ℝⁿ and all α ∈ ℝ.

1. v · (w + u) = v · w + v · u and v · (αw) = α(v · w)   (linearity)

2. v · w = w · v   (symmetry)

3. v · v > 0 for all v ≠ 0  (positivity)

Proof. Using the Einstein summation convention

1. v · (w + u) = v_i (w + u)_i = v_i(w_i + u_i) = v_i w_i + v_i u_i = v · w + v · u

2. v · (αw) = v_i (αw)_i = αv_iw_i = α(v · w)

3. v · w = v_i w_i = w_i v_i = w · v

4. (v, v) = Σ_i v_i² > 0 for v ≠ 0.

Non-orthogonal Axes

Rather than choosing basis vector for a Cartesian coordinate system that are mutually perpendicular (i.e., orthogonal), we can have coordinate basis vectors not mutually perpendicular which may have different length: e_i · e_j ≡ g_ij ≠ δ_ij. We can no longer use the standard formula for dot product Eq. 1.5.2, in this system. Consider a basis {e₁, e₂} such that e₁ ⋅ e₂ ≠ 0, then the dot product between u = (a₁, a₂) and v = (a₁, a₂) is

We can represent the collection of scalar products of the basis vectors as a symmetric matrix, called the metric or the metric tensor:

We can display the metric of a 4D space as a 4 × 4 matrix whose elements are the dot products of the basis vectors:

$$[g]=\left(\begin{array}{ccc}{g}_{00}& g_{01}& \dots \\ g_{10}& g_{11}& \dots \\ ⋮& ⋮& \end{array}\right)=\left(\begin{array}{ccc}{\mathbf{e}}_0\cdot {\mathbf{e}}_0& {\mathbf{e}}_0\cdot {\mathbf{e}}_1& \dots \\ {\mathbf{e}}_1\cdot {\mathbf{e}}_0& {\mathbf{e}}_1\cdot {\mathbf{e}}_1& \dots \\ ⋮& ⋮& \end{array}\right)$$

Thus the diagonal elements are the (squared) magnitudes of the basis vectors, |e₀|², |e₁|², etc., while the off-diagonal elements represent their deviations from orthogonality. Any set of mutually perpendicular bases would be represented by a diagonal metric matrix.

Then the dot product could be expressed in the form

where we employed Einstein summation convention.

«Basis Index Dual Space»