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.

Using a scalar triple product formula, we combine the cross product of two of the vectors and the dot product of one of the vectors. We can write it as follows: abc= (a x b).c. This formula indicates the volume of a parallelepiped with three coterminous edges, for example, a, b, and c.

2025年3月5日 · The scalar triple product of three vectors , , and is denoted and defined by. where denotes a dot product, denotes a cross product, denotes a determinant, and , , and are components of the vectors , , and , respectively. The scalar triple product is a pseudoscalar (i.e., it reverses sign under inversion).

Scalar triple product is the dot product of a vector with the cross product of two other vectors, i.e., if a, b, c are three vectors, then their scalar triple product is a · (b × c). It is also commonly known as the triple scalar product, box product, and mixed product.

[a b c ] = ( a × b) . c. The following conclusions can be drawn, by looking into the above formula: i) The resultant is always a scalar quantity. ii) Cross product of the vectors is calculated first, followed by the dot product which gives the scalar triple product.

The scalar triple product of three vectors $\vc{a}$, $\vc{b}$, and $\vc{c}$ is $(\vc{a} \times \vc{b}) \cdot \vc{c}$. It is a scalar product because, just like the dot product, it evaluates to a single number. (In this way, it is unlike the cross product, which is a vector.)

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