Vector Multiplication

There are several ways to conceptualize vector multiplication.

Dot Product

Two vectors of equal dimensions can be multiplied as a dot product. The notation is a ⋅ b.

It is also known as a scalar product because the multiplication yields a single scalar.

Generally, given two vectors (a and b) with n dimensions, the dot product is computed as:

Concretely, if a and b have three dimensions (labeled x, y, and z), the dot product can be computed as:

julia> using LinearAlgebra

julia> # type '\cdot' and tab-complete into '⋅'
julia> [2,3,4] ⋅ [5,6,7] 
56

Properties

Dot product multiplication is commutative.

The dot product is 0 only when a and b are orthogonal.

The dot product effectively measures how similar the vectors are.

Usage

The dot product is also known as the projection product. The dot product of a and b is equivalent to multiplying the distance of a by the distance of the projection of b into C(a), the column space of a. (Because a vector is clearly of rank 1, this space is in R¹ and forms a line.) Trigonometrically, this is ||a|| ||b|| cos(θ).

This provides a geometric intuition for why the dot product is 0 when a and b are orthogonal: there is no possible projection, and necessarily multiplying by 0 results in 0.

Inner Product

The inner product is a generalization of the dot product. Specifically, the dot product is the inner product in Euclidean space. The notation is ⟨a, b⟩.

Cross Product

Two vectors of 3-dimensional vectors can be multiplied as a cross product. The notation is a × b.

Properties

Cross product multiplication is anti-commutative: a × b = -b × a.

The cross product effectively measures how dissimilar the vectors are.

Usage

The cross product is a vector that is orthogonal to both a and b.

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