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The scalar triple product ... This is known as triple product expansion, or Lagrange's formula, [2] [3] although the latter name is also used for several other formulas.
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.
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 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 ...
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.
denotes the scalar triple product of the three vectors and denotes the scalar product. Care must be taken here to avoid negative or incorrect solid angles. One source of potential errors is that the scalar triple product can be negative if a, b, c have the wrong winding. Computing the absolute value is a sufficient solution since no other ...
However, the above geometry may be used to give an independent proof of the sine rule. The scalar triple product, OA → · (OB → × OC →) evaluates to sin b sin c sin A in the basis shown. Similarly, in a basis oriented with the z-axis along OB →, the triple product OB → · (OC → × OA →), evaluates to sin c sin a sin B.
The formula is valid for all index values, and for any n (when n = 0 or n = 1, ... If c = (c 1, c 2, c 3) is a third vector, then the triple scalar product equals ...