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The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation
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.
The Jacobi triple product identity is the Macdonald identity for the affine root system of type A 1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. Properties [ edit ]
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.
The scalar triple product of three vectors is defined as = = (). Its value is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.
If c = (c 1, c 2, c 3) is a third vector, then the triple scalar product equals =. From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any pair of arguments. For example,
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The scalar coefficient is the triple product of the three vectors. The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product u × v can be interpreted as a vector which is perpendicular to both u and v and whose magnitude is equal to the area of the parallelogram ...