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If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume. If any two vectors in the scalar triple product are equal, then its value is 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.
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 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,
The scalar triple product (also called the box product or mixed triple product) is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by (a b c) and defined as:
A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a ...
Combining the above diagrams for the cross product and the dot product, one can read off the three leftmost diagrams as precisely the three leftmost scalar triple products in the above identity. It can also be shown that the rightmost diagram represents det[u v w]. The scalar triple product identity follows because each is a different ...
Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables = + = + Substitute dy into dx = [() + ()] + By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero () + = Subtracting the second term and multiplying by its inverse gives the triple ...