We're given
[tex]\vec A = -(8.0\,\mathrm m)\,\vec\imath[/tex]
and
[tex]\vec A \times \vec B = (16.0\,\mathrm m^2) \, \vec k[/tex]
Recall that
[tex]\vec\imath \times \vec\jmath = \vec k[/tex]
so [tex]\vec B[/tex] must pointing in the -y direction, which means
[tex]\vec B = -\|\vec B\|\,\vec\jmath[/tex]
Then it follows that
[tex]\vec A \times \vec B = (8.0\,\mathrm m)\|\vec B\| (\vec\imath\times\vec\jmath) = (16.0\,\mathrm m^2) \, \vec k \\\\ \implies \|\vec B\| = \boxed{2.0\,\mathrm m}[/tex]
