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Question
Prove that the following vectors are coplanar:
\[\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}\]
Solution
Given the vectors \[P\left( \hat{i} + \hat{j} + \hat{k} \right), Q\left( 2\hat{i} +3 \hat{j} - \hat{k} \right)\] and \[R\left( - \hat{i} - 2\hat{j} +2 \hat{k} \right)\]
We know the three vectors are coplanar if one of them is expressible as a linear combination of the other two.
Let, \[\hat{i} + \hat{j} + \hat{k} = x \left( 2 \hat{i} + 3 \hat{j} - \hat{k} \right) + y \left( - \hat{i} - 2 \hat{j} + 2 \hat{k} \right) . \]
\[ = \hat{i} \left( 2x - y \right) + \hat{j} \left( 3x - 2y \right) + \hat{k} \left( - x + 2y \right) .\]
\[\Rightarrow 2x - y = 1 , 3x - 2y = 1, - x + 2y = 1\] [Equating the coefficients of \[\hat{i} , \hat{j} , \hat{k}\] respectively]
Solving first two of these equation, we get \[x = 1 , y = 1\]. Clearly these two values satisfy the third equation.
Hence, the given vectors are coplanar.
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