Differences between revisions 3 and 4
⇤ ← Revision 3 as of 2025-03-28 03:59:13 →
Size: 1534
Editor: DominicRicottone
Comment:
← Revision 4 as of 2025-03-28 03:59:53 → ⇥
Size: 1572
Editor: DominicRicottone
Comment: Link
Deletions are marked like this. Additions are marked like this.
Line 45: Line 45:
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 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.)

Vector Multiplication

There are several ways to conceptualize vector multiplication.

Contents

  1. Vector Multiplication
    1. Dot Product
      1. Properties
      2. Applications

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:

dot1.svg

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

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 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 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 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.

CategoryRicottone

LinearAlgebra/VectorMultiplication (last edited 2025-03-28 16:32:43 by DominicRicottone)