How to find a vector given the scalar triple product

Question

Scalar triple product formula: $a (b \times c)$

If I know the value of the product and I know what vectors $b$ and $c$ are, how do I find what vector $a$ is equal to?

Answer 1

Unfortunately, you can't. Your real question is even simpler: if you know $\mathbf{x}$ and $\mathbf{a}\cdot\mathbf{x} = y$, can you solve for $\mathbf{a}$?

The issue is that the dot product is not an invertible operation, so there are lots of ways to choose $\mathbf{x}$ such that $\mathbf{a}\cdot\mathbf{x} = y$. Think about it for the two-dimensional cases: let $\mathbf{a} = [a_1, a_2]$ and $\mathbf{x} = [x_1, x_2]$. The hypothesis is that we know $x_1$, $x_2$, $y$, and that

$$ a_1x_1 + a_2x_2 = y, $$

but this is a single equation with two unknowns ($a_1$ and $a_2$), which has infinitely many solutions. If that seems odd, verify that $\mathbf{a} = [0, \tfrac{y}{x_2}]$ and $\mathbf{a} = [\frac{y}{x_1}, 0]$ are both solutions.