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Wiki/Math/Vector Basics.md
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@@ -37,7 +37,7 @@ Used to project vector $v$ onto $u$
![[VectorProjection.excalidraw]] ![[VectorProjection.excalidraw]]
### Triple Scalar Product ### Triple Scalar Product
Find the Determinant of: Find the Determinant of:
$$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}$$ $$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ v_{1} && v_{2} && v_{3} \\ w_{1} && w_{2} && w_{3} \end{vmatrix}$$
The triple scalar product can also be found by finding the cross product of $v$ and $w$ and then taking the dot product of the resulting vector and $u$. The triple scalar product can also be found by finding the cross product of $v$ and $w$ and then taking the dot product of the resulting vector and $u$.
$$u * (v \times w) = (u \times v) * w $$ $$u * (v \times w) = (u \times v) * w $$
The triple scalar product is equal to the volume of the parallelpiped where each vector represents one adjacent edge. The triple scalar product is equal to the volume of the parallelpiped where each vector represents one adjacent edge.