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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 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 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)''. |
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| This provides a geometric intuition for why the dot product is 0 when ''a'' and ''b'' are orthogonal: there is no possible projection. | 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. |
Vector Multiplication
There are several ways to conceptualize vector multiplication.
Contents
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.
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.