Triple Dot Product

The term triple dot product has distinct meanings in vector and tensor notation. An ordinary dot product is a binary operation that maps two vectors to a scalar, so a dot product of three vectors is not defined by associating the symbols as : this would attempt to take the dot product of a scalar and a vector.

In vector algebra, "triple dot product" is sometimes used informally for the scalar triple product

which uses one cross product and one dot product, not three dot products.

In tensor analysis, a triple dot denotes contraction over three pairs of indices. For third-rank tensors and , one convention gives the scalar

This is a tensor contraction of two tensors, not a product of three vectors.