Vector Operations

Vector operations can be expressed numerically or geometrically.

Contents

  1. Vector Operations
    1. Addition
      1. Properties
    2. Scalar Multiplication
    3. Dot Product
      1. Properties
    4. Inner Product
    5. Cross Product
      1. Properties

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.

add.png

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

Multiplying a vector by a scalar is equivalent to multiplying each component of the vector by the scalar.

Geometrically, scalar multiplication is scaling.

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 aTb.

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

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.

The dot product can be used to extract components of a vector. For example, to extract the X component of a vector a⃗ in R3, take the dot product of it by the unit vector î.

Properties

Dot product multiplication is commutative.

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

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 a⃗ × b⃗ and it is calculated as the determinant of the two vectors together with a vector of [î ĵ k̂] (referring to the unit vectors):

cross.svg

Recall that the determinant of a matrix does not change with transposition, so this 3 by 3 matrix can be constructed either of columns or rows.

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

Properties

Cross product multiplication is anti-commutative: a⃗ × b⃗ = -b⃗ × a⃗.

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