Scalar and Vector Triple Products A scalar triple product is a scalar value that can be calculated for any three vectors in Euclidean space. The scalar...
Scalar and Vector Triple Products
A scalar triple product is a scalar value that can be calculated for any three vectors in Euclidean space. The scalar triple product is defined as the dot product of the vectors corresponding to the three positions in the Cartesian coordinate system.
where (a_i, b_i, c_i) are the coordinates of the vectors (A, B, C) in the Cartesian coordinate system.
Similarly, a vector triple product is a linear form that maps a vector space to a scalar. In other words, it is a function that takes a vector and outputs a scalar. The vector triple product is defined as the dot product of the vectors corresponding to the three positions in the Cartesian coordinate system.
where (a_i, b_i, c_i) are the components of the vectors (A, B, C) in the Cartesian coordinate system.
The scalar triple product and the vector triple product are both examples of scalar multiples, which are linear forms that take a vector and output a scalar. Scalars and vectors are linear combinations of other scalars and vectors, respectively. Scalar multiples are therefore a generalization of scalar products