Question

\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{ and } \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]

If c₂ = −1 and c₃ = 1, show that no value of c₁ can make \[\vec{a,} \vec{b}\text { and } \vec{c}\] coplanar.

Answer 1

\[\text { If } c_2 = - 1 \text { and } c_3 = 1,\text{ then } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat{ i} \text { and } \hat {c} = c_1 \hat {i} - \hat {j} + \hat{k .} \]

\[\text { We know that vectors } \vec{a} , \vec{b} , \vec{c}\text {  are coplanar iff } \left[ \vec{a} \vec{b} c \right] = 0 . \]

\[\text { For } \vec{a} , \vec{b} , \vec{c}\text {  to be coplanar }: \]

\[ \Rightarrow \left[ \vec{a} \vec{b} c \right] = 0\]

\[ \Rightarrow \begin{vmatrix}1 & 1 & 1 \\ 1 & 0 & 0 \\ c_1 & - 1 & 1\end{vmatrix} = 0 \]

\[ \Rightarrow 1\left( 0 - 0 \right) - 1\left( 1 - 0 \right) + 1\left( - 1 - 0 \right) = 0\]

\[ \Rightarrow - 1 - 1 = 0\]

\[ \Rightarrow - 2 = 0\]

\[\text { But this is never possible, whatever be the value of }c_1 . \text { Thus, no vaue of } c_1 \text { can make } \vec{a} , \vec{b} \text { and } \vec{c} \text { coplanar} .\]