Scalar triple product

The scalar triple product of three vectors $\vec{ \mathbf{v} }$, $\vec{ \mathbf{u} }$, and $\vec{ \mathbf{w} }$ is the dot product of one vector with the cross product of the other two. Technically, it does not produce a scalar but a pseudoscalar, as reversing the direction of one of the vectors will change the sign of the scalar.

Properties

• A parallelepiped whose edges are formed by the vectors $\vec{ \mathbf{v} }$, $\vec{ \mathbf{u} }$, and $\vec{ \mathbf{w} }$ will have a volume equal to the scalar triple product of the three vectors.

• ${\vec {\mathbf {v} }}\,\times \,{\vec {\mathbf {w} }}\,\cdot \,{\vec {\mathbf {u} }}={\begin{vmatrix}\mathbf {v} _{x}&\mathbf {w} _{x}&\mathbf {u} _{x}\\\mathbf {v} _{y}&\mathbf {w} _{y}&\mathbf {u} _{y}\\\mathbf {v} _{z}&\mathbf {w} _{z}&\mathbf {u} _{z}\\\end{vmatrix}}$

• ${\vec {\mathbf {v} }}\,\times \,{\vec {\mathbf {w} }}\,\cdot \,{\vec {\mathbf {u} }}={\vec {\mathbf {v} }}\,\cdot \,({\vec {\mathbf {w} }}\,\times \,{\vec {\mathbf {u} }})$

• ${\vec {\mathbf {v} }}\,\times \,{\vec {\mathbf {w} }}\,\cdot \,{\vec {\mathbf {u} }}={\vec {\mathbf {w} }}\,\times \,{\vec {\mathbf {u} }}\,\cdot \,{\vec {\mathbf {v} }}={\vec {\mathbf {u} }}\,\times \,{\vec {\mathbf {v} }}\,\cdot \,{\vec {\mathbf {w} }}$

• If any of the vectors are identical, the scalar triple product will be zero.

See also

• Dot product
• Cross product
• Vector triple product
• Parallelepiped