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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.