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| The dot product is also known as the '''projection product'''. The dot product of ''a'' and ''b'' is equivalent to the [[LinearAlgebra/Projections#Vectors|projection]] of ''b'' onto the column space of ''a'', possibly notated as ''C(a)'', a.k.a. the space formed by all linear combinations of ''a''. (Because a vector is clearly of [[LinearAlgebra/Rank|rank]] 1, this space is in ''R^1^'' and forms a line.) | The dot product is also known as the '''projection product'''. The dot product of ''a'' and ''b'' is equivalent to 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.) |
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.
Applications
The dot product is also known as the projection product. The dot product of a and b is equivalent to 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.)
This provides a geometric intuition for why the dot product is 0 when a and b are orthogonal: there is no possible projection.