anudoneddbv

2022-07-20

How to find a vector given the scalar triple product
Scalar triple product formula: $a\left(b×c\right)$
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?

ri1men4dp

Expert

Unfortunately, you can't. Your real question is even simpler: if you know x and $\mathbf{a}\cdot \mathbf{x}=y$, can you solve for a?
The issue is that the dot product is not an invertible operation, so there are lots of ways to choose x such that $\mathbf{a}\cdot \mathbf{x}=y$. Think about it for the two-dimensional cases: let $\mathbf{a}=\left[{a}_{1},{a}_{2}\right]$ and $\mathbf{x}=\left[{x}_{1},{x}_{2}\right]$. The hypothesis is that we know ${x}_{1}$, ${x}_{2}$, $y$, and that
${a}_{1}{x}_{1}+{a}_{2}{x}_{2}=y,$
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 $\mathbf{a}=\left[0,\frac{y}{{x}_{2}}\right]$ and $\mathbf{a}=\left[\frac{y}{{x}_{1}},0\right]$ are both solutions

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