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Question
Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
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Solution
\[\text{ Necessary Condition: First let }\vec{a} , \vec{b} , \vec{c} \text{ be three coplanar vectors . Then one of them is expressable as a linear combination of the other two . }\]
\[\text{Let }\vec{c} = x \vec{a} + y \vec{b}\text{ for some scalars x, y . }\]
\[\text{Then, }\vec{c} = x \vec{a} + y \vec{b}\text{ for some scalars x, y }\]
\[ \Rightarrow l \vec{a} + m \vec{b} + n \vec{c} = 0,\text{ where }l = x, m = y, n = - 1\]
\[\text{ Thus, if }\vec{a} , \vec{b} , \vec{c}\text{ are coplanar vectors, then there exists scalars l, m, n such that l }\vec{a} + m \vec{b} + n \vec{c} = 0\text{ where l, m, n are all non zero simultaneously . }\]
\[\text{ Sufficient Condition: Let }\vec{a} , \vec{b} , \vec{c}\text{ be three vectors such that there exists scalars l, m, n not all zero simulataneously }\]
\[\text{ satisfying l }\vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\text{ We have tp prove that }\vec{a} , \vec{b} , \vec{c}\text{ are coplanar vectors . }\]
\[\text{Now, l }\vec{a} + m \vec{b} + n \vec{c} = \vec{0} \]
\[ \Rightarrow n \vec{c} = - l \vec{a} - m \vec{b} \]
\[ \Rightarrow \vec{c} = \left( - \frac{1}{n} \right) \vec{a} + \left( - \frac{m}{n} \right) \vec{b} \]
\[ \Rightarrow \vec{c}\text{ is a linear combination of }\vec{a} \text{ and }\vec{b} . \]
\[\text{ Hence }\vec{a} , \vec{b} , \vec{c}\text{ are coplanar vectors }.\]
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