Parameterization part of a problem

Question

How was the parameterization for this problem done? This is the only part of this question that I don't understand. I'd very much appreciate it if you can show/explain the steps taken to get this parameterization.

Answer 1

If $\mathbf{w}_1$, $\mathbf{w}_2$ are two linearly independent vectors, then $$\mathbf{r}(u,v)=u\mathbf{w}_1+v\mathbf{w}_2,\qquad 0\leq u,v\leq 1$$ paremetrize the parallelogram generated by $\mathbf{w}_1$, $\mathbf{w}_2$ whose vertices are $\mathbf{0}$, $\mathbf{w}_1$, $\mathbf{w}_2$, $\mathbf{w}_1+\mathbf{w}_2$. In your example take $\mathbf{w}_1=(3,2,2)$ and $\mathbf{w}_2=(3,3,4)$.