If a,b,c and $\mathrm{A},\mathrm{B},\mathrm{C}\in \mathrm{R}-\left\{0\right\}$such that 

 $\mathrm{aA}+\mathrm{bB}+\mathrm{cC} +\sqrt{\left({\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\right)\left({\mathrm{A}}^2+{\mathrm{B}}^2+{\mathrm{C}}^2\right)}=0$

 then value of $\frac{\mathrm{aB}}{\mathrm{bA}}+\frac{\mathrm{bC}}{\mathrm{cB}}+\frac{\mathrm{cA}}{\mathrm{aC}}$is 

$$\begin{gathered} {\overline{r}}_1=a\hat{i}+b\hat{j}+c\hat{k} and {\overline{r}}_2=A\hat{i}+B\hat{j}+C\hat{k} \\ {\overline{r}}_1.{\overline{r}}_2=aA+bB+cC \\ \left|{\overline{r}}_1\right|\left|{\overline{r}}_2\right|=\sqrt{a^2+b^2+c^2}\sqrt{A^2+B^2+C^2} \\ {\overline{r}}_{1,}{\overline{r}}_2 are anti parallel \\ \Rightarrow \frac{a}{A}=\frac{b}{B}=\frac{c}{C}=k, where k is any cons\tan t \\ \Rightarrow \frac{aB}{bA}+\frac{bC}{cB}+\frac{cA}{aC}=3 \end{gathered}$$