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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 c1 = 1 and c2 = 2, find c3 which makes \[\vec{a,} \vec{b} \text { and } \vec{c}\] coplanar.
Solution
\[\text { If } c_1 = 1 \text{ and } c_2 = 2, \text { then } \vec{a} = \hat{i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text {and} \hat {c} = \hat {i} + 2 \hat {j} + c_3 \hat {k} . \]
\[\text {We know that vectors } \vec{a} , \vec{b} , \vec{c} \text {are coplanar iff } \left[ \vec{a} \vec{b} c \right] = 0 . \]
\[\text { It is given that } \vec{a} , \vec{b} , \vec{c} \text {are coplanar } . \]
\[ \therefore \left[ \vec{a} \vec{b} c \right] = 0\]
\[ \Rightarrow \begin{vmatrix}1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 2 & c_3\end{vmatrix} = 0 \]
\[ \Rightarrow 1\left( 0 - 0 \right) - 1\left( c_3 - o \right) + 1\left( 2 - 0 \right) = 0\]
\[ \Rightarrow - c_3 + 2 = 0\]
\[ \Rightarrow c_3 = 2\]
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