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Vectors, Pure and Applied by T. W. Körner

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4

The secret life of determinants

4.1 The area of a parallelogram

Let us investigate the behaviour of the area images(a, b) of the parallelogram with vertices 0, a, a + b and b.

Our first observation is that it appears that

images

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)

and so

A similar argument shows that

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