<s>[INST] Scalar triple product in terms of scalar products only I am trying to express the scalar triple product $\bf a \cdot (b \times c)$ only in terms of scalar products $\bf a \cdot b$, $\bf b \cdot c$, $\bf c \cdot a$ and the lengths of vectors $a$, $b$ and $c$. I start by writing $$\mathbf{a} = B\, \mathbf{b} +  C\, \mathbf{c} + D\, \mathbf{b \times c} $$ and then performing the dot products with $\bf a, b, c, b \times c$, I arrive at the following equations $$a^2 = B \, (\mathbf{a \cdot b}) + C \, (\mathbf{c \cdot a}) + D \, (\mathbf{a \cdot (b\times c)})$$ $$\mathbf{a \cdot b} = B\, b^2 + C\, (\mathbf{b \cdot c})$$ $$\mathbf{c \cdot a} = B\, (\mathbf{b \cdot c})+ C\, c^2 $$ $$\mathbf{a \cdot (b\times c)} = D\, \mathbf{(b\times c)}^2$$
Unknowns $B$ and $C$ can be easily eliminated, but when eliminating $D$, I necessarily have to square $\bf a \cdot (b \times c)$ to get the final solution $$[\mathbf{a \cdot (b \times c)}]^2 = a^2b^2c^2 - a^2(\mathbf{b \cdot c}) - b^2(\mathbf{c \cdot a}) - c^2(\mathbf{a \cdot b}) + 2(\mathbf{a \cdot b})(\mathbf{b \cdot c})(\mathbf{c \cdot a})$$
Is there a way of getting a similar expression for $\mathbf{a \cdot (b \times c)}$ directly? I would like to see an expression which clearly demonstrates the cyclic property of the scalar triple product. The square in my expression ruins it...
 [/INST] A "similar" expression is the determinant representation
$$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\operatorname{det} S,\qquad S=\left(
\begin{array}{ccc}
a_x & a_y & a_z \\
b_x & b_y & b_z \\
c_x & c_y & c_z
\end{array}\right).$$
The expression that you have obtained can be easily deduced from this one:
$$\Bigl(\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\Bigr)^2=\operatorname{det} SS^T=
\operatorname{det}\left(\begin{array}{ccc}
\mathbf{a}\cdot\mathbf{a} & \mathbf{a}\cdot\mathbf{b} & \mathbf{a}\cdot\mathbf{c} \\
\mathbf{b}\cdot\mathbf{a} & \mathbf{b}\cdot\mathbf{b} & \mathbf{b}\cdot\mathbf{c} \\
\mathbf{c}\cdot\mathbf{a} & \mathbf{c}\cdot\mathbf{b} & \mathbf{c}\cdot\mathbf{c}
\end{array}\right).$$
However the non-squared triple product cannot be polynomially expressed in terms of scalar products only.
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