Scalar Triple Product

#Math

$\displaystyle \vec{a}\cdot(\vec{b}\times \vec{c})=b\cdot(c\times \vec{a})=\vec{c}\cdot(\vec{a}\times \vec{b})$

• A vector with magnitude equal to the volume of a parallelepiped formed by the three vectors
• Can be derived by observing a parallelepiped

$$ \vec{a}\cdot(\vec{b}\times \vec{c})=\begin{vmatrix}

a_{x} & a_{y} & a_{z} \\b_{x} & b_{y} & b_{z} \\c_{x} & c_{y} & c_{z}
\end{vmatrix}$$

$\displaystyle \vec{a}\cdot(\vec{b}\times \vec{c})=(\vec{a}\times \vec{b})\cdot \vec{c}$

• Follows from top equation

Mnemonic

• The vector ordering are all even parity, so you may just cycle the vectors right or leftward

$\text{Triple Scalar Product}$
$$\vec A\vec B\vec C=\det\begin{bmatrix}
A_x & A_y & A_z\\B_x & B_y & B_z\\C_x & C_y & B_z
\end{bmatrix}
= \vec A \cdot (\vec B \times \vec C) = (\vec A \times \vec B) \cdot \vec C = \text{even permutations of }\vec A \vec B \vec C = -\text{(odd permutations of }\vec A \vec B \vec C)$$