= Vector Operations = '''Vector operations''' can be expressed numerically or geometrically. <> ---- == 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. ---- == Dot Product == 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''. 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,,''. {{{ 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 [[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 [[Calculus/Projection#Scalar_Projection|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'''. The dot product can be used to extract components of a vector. For example, to extract the X component of a vector a⃗ in ''R^3^'', take the dot product of it by the [[Calculus/UnitVector|unit vector]] ''î''. === Properties === Dot product multiplication is commutative. * ''a⃗ ⋅ b⃗ = b⃗ ⋅ a⃗'' * ''a^T^b = b^T^a'' 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. ---- == Cross Product == Two vectors in ''R^3^'' space can be multiplied as a '''cross product'''. The notation is ''a⃗ × b⃗'' and it is calculated as the determinant of the two vectors together with a vector of ''[î ĵ k̂]'' (referring to the [[Calculus/UnitVector|unit vectors]]): {{attachment:cross.svg}} Recall that the determinant of a matrix does not change with [[LinearAlgebra/Transposition|transposition]], so this 3 by 3 matrix can be constructed either of columns or rows. The cross product returns a vector that is orthogonal to both a⃗ and b⃗, and reflects how dissimilar the vectors are. Geometrically, the cross product is ''||a⃗|| ||b⃗|| sin(θ) n̂'' where ''θ'' is the angle formed by the two vectors and ''n̂'' is the unit vector normal to the two vectors. === Properties === Cross product multiplication is '''anti-commutative''': ''a⃗ × b⃗ = -b⃗ × a⃗''. ---- == Outer Product == Vectors of any sizes can be multiplied as an '''outer product'''. In calculus this is commonly notated as ''a⃗ ⊗ b⃗'', while in [[LinearAlgebra|linear algebra]] this is usually written out as ''ab^T^''. If ''a'' is a column of size ''m x 1'' and ''b'' is a row of size ''1 x n'', then the outer product is of size ''m x n''. === Properties === * ''a⃗ ⊗ b⃗ = (b⃗ ⊗ a⃗)^T^'' * ''(a⃗ + b⃗) ⊗ c⃗ = (a⃗ ⊗ c⃗) + (b⃗ ⊗ c⃗)'' * ''c⃗ ⊗ (a⃗ + b⃗) = (c⃗ ⊗ a⃗) + (c⃗ ⊗ b⃗)'' * ''d(a⃗ ⊗ b⃗) = (da⃗) ⊗ b⃗ = a⃗ ⊗ (db⃗)'' * ''(a⃗ ⊗ b⃗) ⊗ c⃗ = a⃗ ⊗ (b⃗ ⊗ c⃗)'' Because every column in an outer product is a linear combination of ''a⃗'', there is always multicolinearity. Therefore an outer product is always of [[LinearAlgebra/Rank|rank]] 1. ---- CategoryRicottone