TITLE: Is every vector in $\mathbb Z^3$ a cross product?
QUESTION [33 upvotes]: Is every $3$-dimensional vector $v$ with integer coordinates a cross product of two other vectors with integer coordinates?

I have written a program to check for $v$ with entries between $-7$ and $7$. Every $v$ that small can be expressed as a cross product of two other vectors with integer coordinates.
But I can't come up with a general proof.
Apart from the empirical evidence from my experiment on small $v$, another reason to think this is true is that it's almost enough to find two small independent vectors, $u$ and $w$, with integer coordinates that are perpendicular to $v$. Playing around with integer relation algorithms has taught me that such $u$ and $w$ should be plentiful. The cross product of $u$ and $w$ is a scalar multiple of $v$ - call it $kv$. $k$ is an integer; in most cases $|k| = 1$. If it isn't, pick a different $u$ and $w$.
A similar but much easier question was this: Is every vector in $\Bbb R^3$ a cross product?.
Note: The answer given here was used to solve Diophantine equations so the question is about number theory.

REPLY [37 votes]: Let us write $v=(a, b, c)$, and consider the three vectors
$$ w_1=(0, c, -b), \quad w_2=(-c,0,a), \qquad w_3=(b, -a, 0). $$
Note that $w_1 \times w_2=cv$, $w_1 \times w_3=-bv$. This way, let $d=\gcd(b, c)$, and write it as $d=\lambda b+\mu c$, so that
$$ \frac{w_1}{d} \times\left( \mu w_2-\lambda w_3 \right)=v. $$
Here $\mu w_2-\lambda w_3$ and $w_1/d$ have integer coordinates, as we can easily check.