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In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors.The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
2.3 Product rule for multiplication by a scalar. 2.4 Quotient rule for division by a scalar. 2.5 Chain rule. 2.6 Dot product rule. ... When the Laplacian is equal to ...
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 ...
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 (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 ...
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,
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 ...