The secret life of determinants
4.1 The area of a parallelogram
Let us investigate the behaviour of the area (a, b) of the parallelogram with vertices 0, a, a + b and b.
Our first observation is that it appears that
To see this, examine Figure 4.1 showing the parallelogram OAXB having vertices O at 0, A at a, X at a + b and B at b together with the parallelogram OAX′B′ having vertices O at 0, A at a, X′ at (1 + λ)a + b and B′ at λa + b. Looking at the diagram, we see that (using congruent triangles)
A similar argument shows that