The Common Sense of the Exact Sciences

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Similarly b' a" equals h a' times the area A b ; but the area A B itself is a^. Thus we see that the identity P . o'^Q = a' b" — b' a" 204 THE COMMON SENSE OP THE EXACT SCIENCES, may be read {aa + 6y3) (a'o + h' ^) = {aV - I a') a^.
Or, the determinant is equal to tlie parallelogram on the alternate units magnified in the ratio of 1 to ab' — h a'. It obviously vanishes it ah' —ha' = 0, or if ajb = a'jV. Iq this case p and q lie, by the property of similar triangles, on the same straight line th
...rough o, and therefore, as we should expect, the determinant p . o Q is zero.
The reader will find little diflBculty in discovering like properties for a determinant formed from three alternate units. In this case there will be a geometrical relation between certain volumes, which may be ob- tained by stretches in the manner explained on p. 139.' We have in this section arrived at a legitimate interpretation of the product of two directed steps or vectors. We find that their product is an area^ or ac- cording to our previous convention (see p.


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