Each of x, y, z, a, b and c is a positive real number that satisfy:

(ay-bx)/c = (cx-az)/b = (bz-cy)/a

Determine with proof, the ratio x:y:z in terms of a, b and c.

(ay – bx)/c = (cx – az)/b = (bz – cy)/a  =  k  say             (1)

Define vectors u and v, in terms of their three components, as follows:

            u = (a, b, c)                   v = (x, y, z)      

Then the vector product,    u x v = (bz – cy, cx – az, ay – bx)
                                                = (ka, kb, kc)    using (1)
                                                = k u

This implies that  u x v  is parallel to u, which is a contradiction for non-zero u and v, unless u and v are themselves parallel (k = 0).

Now,     u, v parallel      =>        x : y : z  =  a : b : c

Posted by Harry on 2012-11-27 14:04:33