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A quaternion of the form a + 0 i + 0 j + 0 k, where a is a real number, is called scalar, and a quaternion of the form 0 + b i + c j + d k, where b, c, and d are real numbers, and at least one of b, c, or d is nonzero, is called a vector quaternion. If a + b i + c j + d k is any quaternion, then a is called its scalar part and b i + c j + d k ...
One possible multiplication table is described in the Multiplication table section, but it is not unique. [5] 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.
Multiplication table of the quaternion group as a subgroup of SL(2,3). The field elements are denoted 0, +, −. There is also an important action of Q 8 on the 2-dimensional vector space over the finite field = {,,} (table at right).
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 ...
Euclidean vectors such as (2, 3, 4) or (a x, a y, a z) can be rewritten as 2 i + 3 j + 4 k or a x i + a y j + a z k, where i, j, k are unit vectors representing the three Cartesian axes (traditionally x, y, z), and also obey the multiplication rules of the fundamental quaternion units by interpreting the Euclidean vector (a x, a y, a z) as 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.
If the cross product of two vectors is the zero vector (that is, a × b = 0), then either one or both of the inputs is the zero vector, (a = 0 or b = 0) or else they are parallel or antiparallel (a ∥ b) so that the sine of the angle between them is zero (θ = 0° or θ = 180° and sin θ = 0). The self cross product of a vector is the zero ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.