Question

Prove that the following vectors are coplanar:
\[2 \hat{i} - \hat{j} + \hat{k} , \hat{i} - 3 \hat{j} - 5 \hat{k} \text{ and }3 \hat{i} - 4 \hat{j} - 4 \hat{k}\]

Answer 1

Given the vectors

\[P\left( 2 \hat{i} - \hat{j} + \hat{k} \right), Q\left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right)\] and \[R\left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)\] 
 We know the three vectors are coplanar if one of them is expressible as a linear combination of the other two.
Let, \[2 \hat{i} - \hat{j} + \hat{k} = x \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right) + y \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right) . \]
\[ = \hat{i} \left( x + 3y \right) + \hat{j} \left( - 3x - 4y \right) + \hat{k} \left( - 5x - 4y \right) .\]

\[\Rightarrow x + 3y = 2, - 3x - 4y = - 1, - 5x - 4y = 1\]                            [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.