已知x^2+y^2=3,z^2+w^2=3,xz+yw=0,求xy+zw的值

Question

已知x^2+y^2=3,z^2+w^2=3,xz+yw=0,求xy+zw的值

Answer 1

因为xz+yw=0,所以yw=-xz,所以y=-xz/w
所以x^2+y^2=3即x^2+x^2*z^2/w^2=3
即x^2(1+z^2/w^2)=3,x^2[(w^2+z^2)/w^2]=3
因为w^2+z^2=3,所以上式即x^2/w2=1,x^2=w^2,x=w或者x=-w,
又因为x^2+y^2=3,z^2+w^2=3,所以y^2=z^2,y=z或者y=-z,
又因为xz+yw=0,所以x=w时,y=-z或者x=-w时,y=z.
所以xy+zw=0,所以xy+zw的值为0.
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