Question

Prove that the following vectors are non-coplanar:

\[3 \hat{i} + \hat{j} - \hat{k} , 2 \hat{i} - \hat{j} + 7 \hat{k}\text{ and }7 \hat{i} - \hat{j} + 23 \hat{k}\]

Answer 1

Let if possible the given vectors are coplanar. Then one of the given vector is expressible in terms of the other two.
We have,
\[3 \hat{i} + \hat{j} - \hat{k} = x(2 \hat{i} - \hat{j} + 7 \hat{k} ) + y(7 \hat{i} - \hat{j} + 23 \hat{k} ) . \]
\[ = \hat{i} (2x + 7y) + \hat{j} ( - x - y) + \hat{k} (7x + 23y) . \]
\[ \Rightarrow 2x + 7y = 3 , x + y = - 1, 7x + 23y = - 1 . \]
By solving the first two equations, we get
\[ \Rightarrow x = - 2, y = 1 .\]
Clearly these values of x and y does not satisfy the third equation.
Hence the given vectors are non-coplanar.