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The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule [1] and a magnitude equal to the area of the parallelogram that the vectors span. [2] The cross product is defined by the formula [8] [9]
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
Block matrix operations; Cracovian product, defined as A ∧ B = B T A; Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the entries of the Hadamard product; Hadamard product of two matrices of the same size, resulting in a matrix of the same size, which is the product entry-by-entry
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product b∧c, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction [6] can be used, so the formula becomes [7]
In three dimensions the cross product is invariant under the action of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of x × y under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7).
The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalizing the situation over a field F, where every nonzero element is invertible. [59] Matrices over superrings are called supermatrices. [60]