information:extraction
Triple product

There are two ternary operations involving dot product and cross product.

The scalar triple product of three vectors is defined as a ⋅ (b × c) = b ⋅ (c × a) = c ⋅ (a × b). {\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c})=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a})=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b}).} Its value is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.

The vector triple product is defined by a × (b × c) = (a ⋅ c) b − (a ⋅ b) c. {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c})=(\mathbf {a} \cdot \mathbf {c})\,\mathbf {b} -(\mathbf {a} \cdot \mathbf {b})\,\mathbf {c}.} This identity, also known as Lagrange's formula, may be remembered as "ACB minus ABC", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations in physics.