If $\stackrel{ }{\overline{\mathrm{a}},}\overline{\mathrm{b}} \mathrm{and} \overline{\mathrm{c}}$ are non –coplanar vectors and if $\overline{\mathrm{d}}$ is such that $\overline{\mathrm{d}}=\frac{1}{\mathrm{x}}(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})\text{ and }\overline{\mathrm{a}}=\frac{1}{\mathrm{y}}(\overline{\mathrm{b}}+\overline{\mathrm{c}}+\overline{\mathrm{d}})$ where x and y are non zero real numbers, then $\frac{1}{\mathrm{xy}}(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}+\overline{\mathrm{d}})=$

$\begin{array}{l}\mathrm{Given} \mathrm{x}\overline{\mathrm{d}}=\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\Rightarrow \overline{\mathrm{d}}\stackrel{ }{=\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}}{x}\to \left(1\right) \mathrm{and} }\\ \overline{\mathrm{a}}\mathrm{y}=\overline{\mathrm{b}}+\overline{\mathrm{c}}+\overline{\mathrm{d}}\\ \overline{\mathrm{a}}\mathrm{y}=\overline{\mathrm{b}}+\overline{\mathrm{c}}+\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}}{\mathrm{x}} \\ \Rightarrow \overline{\mathrm{a}}\mathrm{y}-\overline{\mathrm{b}}-\mathrm{c}-\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}}{\mathrm{x}} =0\\ \overline{\mathrm{a}}(y-\frac{1}{x})-\overline{\mathrm{b}}(1+\frac{1}{x})-\overline{\mathrm{c}}(1+\frac{1}{x})=0\\ \sin ce \overline{\mathrm{a}},\overline{\mathrm{b}},\mathrm{c} \mathrm{are} \mathrm{non}-\mathrm{coplanar} \mathrm{then} \mathrm{y}-\frac{1}{\mathrm{x}}=0\Rightarrow \mathrm{xy}=1 \mathrm{and}\\ 1+\frac{1}{x}=0\\ \Rightarrow x=-1 and y=-1\\ now from \left(1\right) \frac{1}{xy}(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}+\overline{d})=0\end{array}$