Vector Operations

Vector operations can be expressed numerically or geometrically.

Addition

Vectors are added numerically as piecewise summation. Given vectors a⃗ and b⃗ equal to [1,2] and [3,1], their sum is [4,3].

Vectors are added geometrically by joining them tip-to-tail, as demonstrated in the below graphic.

Properties

These two views of vector addition also demonstrate that addition is commutative.

Furthermore, it follows that if a⃗ + b⃗ = c⃗, then c⃗ - b⃗ = a⃗.

Scalar Multiplication

Dot Product

Vectors of equal dimensions can be multiplied as a dot product. In calculus this is commonly notated as a⃗ ⋅ b⃗, while in linear algebra this is usually written out as a^Tb.

In R³ space, the dot product can be calculated numerically as a⃗ ⋅ b⃗ = a₁b₁ + a₂b₂ + a₃b₃. More generally this is expressed as Σa_ib_i.

julia> using LinearAlgebra

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

Geometrically, the dot product is ||a⃗|| ||b⃗|| cos(θ) where θ is the angle formed by the two vectors. This demonstrates that dot products reflect both the distance of the vectors and their similarity.

The operation is also known as a scalar product because it yields a single scalar.

Lastly, in terms of linear algebra, a ⋅ b is equivalent to multiplying the distance of a by the scalar projection of b into the column space of a. Because a vector is clearly of rank 1, this column space is in R¹ and forms a line. As a result of this interpretation, this operation is also known as a projection product.

Properties

Dot product multiplication is commutative.

• a⃗ ⋅ b⃗ = b⃗ ⋅ a⃗

• a^Tb = b^Ta

The cos(θ) component of the alternative definition provides several useful properties.

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

• The dot product is positive only when θ is acute.

• The dot product is negative only when θ is obtuse.

The linear algebra view corroborates this: when a and b are orthogonal, there is no possible projection, so the dot product must be 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 in R³ space can be multiplied as a cross product. The notation is a × b.

The cross product is a vector that is orthogonal to both a and b, and reflects how dissimilar the vectors are.

Properties

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

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