JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

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) \[{{\lambda }_{3}}+{{\lambda }_{2}}=2\]

Correct Answer: B

Solution :

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 \[{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4\].