I'm sure this is meant as joshing, but I don't really get it. I reduce abstract concepts to geometry all the time in my classes-- every math person does. For instance on the first day of Calc, I reduce the rate of change of a fcn to the slope of a tangent line.
It is true though that at least in the US many students see determinants for the first time in Calc III where they are used for Jacobians and are introduced as computational gadgets (many instructors would say that they did not have the extra day to describe them in another way, such as the linked-to article does). That's too bad, and I could definitely understand it leaving a bad taste.
In uni we defined determinants starting with systems of linear equations and showing that hey, all these formulas for solving m-by-n systems can be generalised using this one monstrous equation, that has a simple recursive definition. We'll call it a determinant.
The rhombus is in the II or IV quandrant. Alternatively, the endpoint of the vector sum is of the form (-x, y) or (x, -y), where x and y are positive numbers.