If `u=ax+by+cz`, `v=ay+bz+cx`, `w=az+bx+cy`, then the value of `|{:(a,b,c),(b,c,a),(c,a,b):}|xx|{:(x,y,z),(y,z,x),(z,x,y):}|` is

Options

A. A. `u^(2)+v^(2)+w^(2)-2uvw`
B. B. `u^(3)+v^(3)+w^(3)-3uvw`
C. C. `0`
D. D. none of these

Answer

B

Explanation

`(b)` `Delta_(1)Delta_(2)=|{:(ax+by+cz,ay+bz+cx,az+bx+cy),(bx=cy+az,by+cz+ax,bz+cx+ay),(cx+ay+bz,cy+az+bx,cz+ax+by):}|`
`=|{:(u,v,w),(w,u,v),(v,w,u):}|`
`=u(u^(2)-vw)-v(wu-v^(2))+w(w^(2)-uv)`
`=u^(3)+v^(3)+w^(3)-3uvw`