• # question_answer Let a, b, c are three non-coplanar vectors such that  ${{\mathbf{r}}_{1}}=\mathbf{a}-\mathbf{b}+\mathbf{c},\,\,{{\mathbf{r}}_{2}}=\mathbf{b}+\mathbf{c}-\mathbf{a},\,\,{{\mathbf{r}}_{3}}=\mathbf{c}+\mathbf{a}+\mathbf{b},$       $\mathbf{r}=2\mathbf{a}-3\mathbf{b}+4\mathbf{c}.$ If $\mathbf{r}={{\lambda }_{1}}{{\mathbf{r}}_{1}}+{{\lambda }_{2}}{{\mathbf{r}}_{2}}+{{\lambda }_{3}}{{\mathbf{r}}_{3}},$ then A) ${{\lambda }_{1}}=7$ B) ${{\lambda }_{1}}+{{\lambda }_{3}}=3$ C) ${{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4$ D) ${{\lambda }_{3}}+{{\lambda }_{2}}=2$

• We have $\mathbf{r}={{\lambda }_{1}}{{\mathbf{r}}_{1}}+{{\lambda }_{2}}{{\mathbf{r}}_{2}}+{{\lambda }_{3}}{{\mathbf{r}}_{3}}$
• $\Rightarrow 2\mathbf{a}-3\mathbf{b}+4\mathbf{c}=({{\lambda }_{1}}-{{\lambda }_{2}}+{{\lambda }_{3}})\mathbf{a}$
• $+(-{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}})\mathbf{b}+({{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}})\mathbf{c}$                    $\Rightarrow {{\lambda }_{1}}-{{\lambda }_{2}}+{{\lambda }_{3}}=2,-{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=-3,{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4$ $(\because \,\,\,\mathbf{a},\,\mathbf{b},\,\mathbf{c}$ are non-coplanar)
• $\Rightarrow {{\lambda }_{1}}=\frac{7}{2},$ ${{\lambda }_{2}}=1,$ ${{\lambda }_{3}}=-\frac{1}{2}$
• Therefore, ${{\lambda }_{1}}+{{\lambda }_{3}}=3$ and ${{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4$.