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.

The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.

Geometrically, the scalar triple product

is the (signed) volume of the parallelepiped defined by the three vectors given.

Strictly speaking, a scalar does not change at all under a coordinate transformation. (For example, the factor of 2 used for doubling a vector does not change if the vector is in spherical vs. rectangular coordinates.) However, if each vector is transformed by a matrix then the triple product ends up being multiplied by the determinant of the transformation matrix. That is, the triple product of covariant vectors is more properly described as a scalar density.

$${\displaystyle {\begin{aligned}T\mathbf {a} \cdot (T\mathbf {b} \times T\mathbf {c} )&=\det {\begin{pmatrix}T\mathbf {a} &T\mathbf {b} &T\mathbf {c} \end{pmatrix}}\\&=\det \left(T{\begin{pmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{pmatrix}}\right)\\&=\det(T)\det \!{\begin{pmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{pmatrix}}\\&=\det(T)\left(\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\right)\end{aligned}}}$$

Some authors use "pseudoscalar" to describe an object that looks like a scalar but does not transform like one. Because the triple product transforms as a scalar density not as a scalar, it could be called a "pseudoscalar" by this broader definition. However, the triple product is not a "pseudoscalar density".

When a transformation is an orientation-preserving rotation, its determinant is +1 and the triple product is unchanged. When a transformation is an orientation-reversing rotation then its determinant is −1 and the triple product is negated. An arbitrary transformation could have a determinant that is neither +1 nor −1.