A vector c of magnitude 6 is perpendicular to both a and b and makes an acute angle with the z-axis.

Explanation

1. 01

   $\vec{a} \times \vec{b} = -8\hat{i} + 8\hat{j} + 4\hat{k}$ is perpendicular to both, with magnitude 12; the unit vectors along it are $\pm\frac{1}{3}(-2\hat{i} + 2\hat{j} + \hat{k})$.

2. 02

   Magnitude 6 gives $\pm(-4\hat{i} + 4\hat{j} + 2\hat{k})$; an acute angle with the $z$-axis needs a positive $\hat{k}$ component, so $\vec{c} = -4\hat{i} + 4\hat{j} + 2\hat{k}$.