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प्रश्न
\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{ and } \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]
If c2 = −1 and c3 = 1, show that no value of c1 can make \[\vec{a,} \vec{b}\text { and } \vec{c}\] coplanar.
उत्तर
\[\text { If } c_2 = - 1 \text { and } c_3 = 1,\text{ then } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat{ i} \text { and } \hat {c} = c_1 \hat {i} - \hat {j} + \hat{k .} \]
\[\text { We know that vectors } \vec{a} , \vec{b} , \vec{c}\text { are coplanar iff } \left[ \vec{a} \vec{b} c \right] = 0 . \]
\[\text { For } \vec{a} , \vec{b} , \vec{c}\text { to be coplanar }: \]
\[ \Rightarrow \left[ \vec{a} \vec{b} c \right] = 0\]
\[ \Rightarrow \begin{vmatrix}1 & 1 & 1 \\ 1 & 0 & 0 \\ c_1 & - 1 & 1\end{vmatrix} = 0 \]
\[ \Rightarrow 1\left( 0 - 0 \right) - 1\left( 1 - 0 \right) + 1\left( - 1 - 0 \right) = 0\]
\[ \Rightarrow - 1 - 1 = 0\]
\[ \Rightarrow - 2 = 0\]
\[\text { But this is never possible, whatever be the value of }c_1 . \text { Thus, no vaue of } c_1 \text { can make } \vec{a} , \vec{b} \text { and } \vec{c} \text { coplanar} .\]
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