= Vector Multiplication =

There are several ways to conceptualize '''vector multiplication'''.

<<TableOfContents>>

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== 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:

{{attachment:dot1.svg}}

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

{{attachment:dot2.svg}}

{{{
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 [[LinearAlgebra/Orthogonality|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 [[LinearAlgebra/Projections#Vectors|projection]] of ''b'' into ''C(a)'', the column space of ''a''. (Because a vector is clearly of [[LinearAlgebra/Rank|rank]] 1, this space is in ''R^1^'' 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.

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== 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⟩''.

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== 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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CategoryRicottone