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In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: Dot product – also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the ...
Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two quaternions.
Red, green and blue arrows represent multiplication by i, j, and k, respectively. Multiplication by negative numbers is omitted for clarity. Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±j, ±k} forms a group under multiplication.
We can express quaternion multiplication in the modern language of vector cross and dot products (which were actually inspired by the quaternions in the first place [14]). When multiplying the vector/imaginary parts, in place of the rules i 2 = j 2 = k 2 = ijk = −1 we have the quaternion multiplication rule:
When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H: =. In the case of vectors with real components, this definition is the same as in the real case.
Unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product. Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x × y through ...
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.) In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented ...
While he could not "multiply triples", he saw a way to do so for quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge: i 2 = j 2 = k 2 = ijk = −1
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