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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.

See also

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