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| = Vector Multiplication = | = Vector Operations = |
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| There are several ways to conceptualize '''vector multiplication'''. | '''Vector operations''' can be expressed numerically or geometrically. |
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| == 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. {{attachment:add.png||height=200px}} === 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. ---- |
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| Two vectors of equal dimensions can be multiplied as a '''dot product'''. The notation is ''a ⋅ b''. | Vectors of equal dimensions can be multiplied as a '''dot product'''. In calculus this is commonly notated as ''a⃗ ⋅ b⃗'', while in [[LinearAlgebra|linear algebra]] this is usually written out as ''a^T^b''. |
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| 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}} |
In ''R^3^'' space, the dot product can be calculated numerically as ''a⃗ ⋅ b⃗ = a,,1,,b,,1,, + a,,2,,b,,2,, + a,,3,,b,,3,,''. More generally this is expressed as ''Σa,,i,,b,,i,,''. |
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| 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 [[Calculus/Distance|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 [[LinearAlgebra/Projection#Vectors|scalar projection]] of ''b'' into the column space of ''a''. Because a vector is clearly of [[LinearAlgebra/Rank|rank]] 1, this column space is in ''R^1^'' and forms a line. As a result of this interpretation, this operation is ''also'' known as a '''projection product'''. |
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| * ''a⃗ ⋅ b⃗ = b⃗ ⋅ a⃗'' * ''a^T^b = b^T^a'' |
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| The dot product is 0 only when ''a'' and ''b'' are [[LinearAlgebra/Orthogonality|orthogonal]]. | The ''cos(θ)'' component of the alternative definition provides several useful properties. * The dot product is 0 only when ''a'' and ''b'' are [[Calculus/Orthogonality|orthogonal]]. * The dot product is positive only when ''θ'' is acute. * The dot product is negative only when ''θ'' is obtuse. The linear algebra view corroborates this: when ''a'' and ''b'' are orthogonal, there is no possible projection, so the dot product must be 0. ---- |
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| === Applications === | == Inner Product == |
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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 '''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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| This provides a geometric intuition for why the dot product is 0 when ''a'' and ''b'' are orthogonal: there is no possible projection. | ---- == Cross Product == Two vectors in ''R^3^'' space can be multiplied as a '''cross product'''. The notation is ''a × b''. The cross product is a vector that is orthogonal to both ''a'' and ''b'', and reflects how dissimilar the vectors are. === Properties === Cross product multiplication is '''anti-commutative''': ''a × b = -b × a''. |
Vector Operations
Vector operations can be expressed numerically or geometrically.
Contents
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.
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.
Properties
Dot product multiplication is commutative.
a⃗ ⋅ b⃗ = b⃗ ⋅ a⃗
aTb = bTa
The cos(θ) component of the alternative definition provides several useful properties.
The dot product is 0 only when a and b are orthogonal.
The dot product is positive only when θ is acute.
The dot product is negative only when θ is obtuse.
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 a × b.
The cross product is a vector that is orthogonal to both a and b, and reflects how dissimilar the vectors are.
Properties
Cross product multiplication is anti-commutative: a × b = -b × a.