This question is nearly identical - I would have commented there, but I suppose the post is too old. I would like to extend that scenario slightly.
Using Tpofofn's diagram - vector (a,b) is clockwise from vector (c,d) - and hence the determinant (a,b) x (c,d) is positive.
Consider a scenario where the two vectors are moving towards each other. The parallelogram is becoming stretched as the two vectors approach each other. Finally, the two points are colinear and hence the determinant is 0.
Here's my question - When the two lines cross, the determinant (a,b) x (c,d) is now negative. But why? What does the parallelogram look like as the two vectors continue to move away from each other?