$If x\cos \theta = y\cos ( \theta +{120}^0) = z \cos ( \theta +{240}^0) then xy + yz + zx =$

$\begin{array}{rcl}\mathrm{x} \mathrm{cos\theta }& =& \cos (\mathrm{\theta }+{120}^{ })=3\cos (\mathrm{\theta }+{240}^0)=\mathrm{k}\\ & & \frac{\mathrm{k}}{\mathrm{x}}+\frac{\mathrm{k}}{\mathrm{y}}+\frac{\mathrm{k}}{\mathrm{z}}\\ & =& \mathrm{cos\theta }+\cos (\mathrm{\theta }+{120}^0)+\cos (\mathrm{\theta }+{240}^0)\\ & =& \mathrm{cos\theta }+\cos (\mathrm{\theta }+{120}^0)+\cos ({360}^0+(\mathrm{\theta }-{120}^0\left)\right)\\ & =& \mathrm{cos\theta }+\cos (\mathrm{\theta }+{120}^0)+\cos (\mathrm{\theta }-{120}^0)\\ & =& \mathrm{cos\theta }+2\mathrm{cos\theta } \cos {120}^0\\ & =& \mathrm{cos\theta }+2\left(\frac{-1}{2}\right) \mathrm{cos\theta }\\ & =& \mathrm{cos\theta }-\mathrm{cos\theta }\\ & =& 0\\ \mathrm{xy}+\mathrm{yz}+\mathrm{zx}& =& 0\end{array}$