I wanted to prove that if we change order of vectors involved in a scalar triple product in a cyclic fashion , then the product remain same . I want an elegant proof of it involving simple algebra of vectors only .
I have already proved it by showing that the volume of parallelopied remains same after the cyclic shift .
I also proved by considering
A = ai + bj + ck
B = di + ej + fk
C = gi + hj + pk
Now by expanding the scalar triple product by determinants and arranging in both cases, I can easily prove that it remains same . But what I am asking is an more algebraic and elegant way . I mean , without just simply expanding and without geometrical interpretation. I meant , I want a prove involving only vectors like using properties of vectors .
Thanks in advance .
Since $\mathbf a\times \mathbf c$ is orthogonal to $\mathbf a$, we have : $$\mathbf a \cdot (\mathbf b \times \mathbf c) = \mathbf a \cdot (( \mathbf a + \mathbf b)\times \mathbf c)$$
Likewise, we have $(\mathbf a + \mathbf b) \cdot ((\mathbf a + \mathbf b) \times \mathbf c) = 0$. Combining both equations we get : \begin{align} \mathbf a \cdot (\mathbf b \times \mathbf c) &= - \mathbf b \cdot (( \mathbf a +\mathbf b)\times \mathbf c) \\ &= - \mathbf b\cdot( \mathbf a\times \mathbf c) \\ &= \mathbf b\cdot (\mathbf c \times \mathbf a) \end{align}