We have the relation in $R^2$ given by $(( x_1, x_2) R (y_1, y_2)) \iff ((x_1 \geq y_1) \land (x_2 \geq y_2))$ for all $( x_1, x_2), (y_1, y_2)$ in $R^2$.
The professor said it is not complete by giving a counter example. Say $( x_1, x_2) = (1,2)$ and $(y_1, y_2) = (2,1)$.
Why would we say it is not complete when we have that for all $x, y,$ either $xRy$ or $yRx$ or both?
It seems to me that $\geq$ gives us the opportunity to say it is complete. Is this not valid when we have pairs like $( x_1, x_2), (y_1, y_2)$? Can we use any numbers to say this relation is not complete when we already have the conditions $x_1 \geq y_1$ and $x_2 \geq y_2)$ given to us at the beginning?
I might be misunderstanding your question, but here goes.
This relation is defined on $\mathbb{R}^2$, so for it to be a total order (which seems to be the way you use 'complete' here) we would need that for any $\vec{x},\vec{y}\in \mathbb{R}^2$ we have $x R y$ or $y R x$ (not exclusive or). I think we can agree your professor gave two vectors in $\mathbb{R}^2$ we can call $\vec{x}=(1,2)$ and $\vec{y}=(2,1)$. Both of these lie in $\mathbb{R}^2$, so for the relation R to be a total order we need $(1,2) R (2,1)$ or $(2,1) R (1,2)$. But since neither of these are true ($1\leq 2$ but $2\not\leq 1$ in both cases), we know that his vectors are incomparable, meaning the relation can't be a total order.