Question

lf $u=$ $ax$ $+$ $by$ $+$ cz, $v=$ $ay$ $+$ $bz$ $+$ $cx$, $w=az+bx+$ $cy$ then $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|.\left|\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right|$

• A. $u^3+v^3+w^{3}3uvw$
• B. $3uvw{u}^3-v^3-w^3$
• C. $u+v+w$
• D. $u^2+v^2+w^2-uv-vw-wu$

Answer 1

The correct option is A $u^3+v^3+w^{3}3uvw$

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|\cdot \left|\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right|=\left|\begin{array}{ccc}ax+by+cz& ay+bz+cx& az+bx+cy\\ bx+cz+az& by+cz+ax& bz+cx+ay\\ cx+ay+bz& cy+az+bx& cz+ax+by\end{array}\right|$
$=\left|\begin{array}{ccc}u& v& w\\ w& u& v\\ v& w& u\end{array}\right|$
$=u(u^2-vw)-v(wu-v^2)+w(w^2-uv)$

$=u^3+v^3+w^3-3uvw$