$\begin{aligned} & \vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 3 & 2 & x \\ 1 & -1 & 1\end{array}\right|=(2+x) \hat{i}+(x-3) \hat{j}+(-3-2) \hat{k} \\ & \text { now } r=|\vec{a} \times \vec{b}|=\sqrt{(2+x)^2+(x-3)^2+(-5)^2} \\ & =\sqrt{2 x^2-2 x+38} \\ & =\sqrt{2\left(x-\frac{1}{2}\right)^2+\frac{75}{2}} \\ & \Rightarrow r_{\min }=\sqrt{\frac{75}{2}} \text { at } x=\frac{1}{2} \\ & \Rightarrow r \geq 5 \cdot \sqrt{\frac{3}{2}}\end{aligned}$