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The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and ...
A cross product, that is a vector-valued, bilinear, anticommutative and orthogonal product of two vectors, is defined in seven dimensions. Along with the more usual cross product in three dimensions it is the only such product, except for trivial products.
The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
Seven-dimensional cross product; T. Tensor product; Triple product; W. Wedge product This page was last edited on 17 December 2020, at 23:46 ...
However, the cross product in 7 dimensions does not share all the properties of the cross product in 3 dimensions. For example, the direction of a × b in 7-dimensions may be the same as c × d even though c and d are linearly independent of a and b. Also the seven-dimensional cross product is not compatible with the Jacobi identity. [9]
Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic.
That source is about the cross product in 3D: it deduces the Schwarz inequality in general, then shows that in 3D the result gives the expression above. Nowhere does it work out a 7D result, so it is irrelevant to this article which is on the seven-dimensional cross product.--JohnBlackburne words deeds 12:51, 18 April 2010 (UTC)
The geometric reason for only two cross products in 3D relates to the number of distinct unit 3-forms in 3D. From Seven-dimensional cross product#Using geometric algebra, one can see that the cross product depends on the choice of