Is it possible to solve the equation $x \times b + y \times c + z \times d = v$ for $b, c, d$?

Question

I have four fixed vectors $\vec x$ , $\vec y$ , $\vec z $ and $\vec v$

is it then possible to find $\vec b$, $\vec c$ and $\vec d$ with the equation:

$\vec x \times \vec b + \vec y \times \vec c + \vec z \times \vec d = \vec v$

x = [10, 0, 0] y = [10, 0, 10] z = [0, 0, 10] v = [0, 0, 230]

(these are just values i picked randomly)

Answer 1

If you think of it in terms of components, you have three equations (one for each dimension) and nine unknowns (three components each for $b,c,d$.) You are short six equations for a unique solution. Finding some solution should be quite possible unless there are dependencies in your vectors.