Question

Prove that the following vectors are non-coplanar:

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

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,
\[\hat{i} + 2 \hat{j} + 3 \hat{k} = x(2 \hat{i} + \hat{j} + 3 \hat{k} ) + y( \hat{i} + \hat{j} + \hat{k} ) . \]
\[ = \hat{i} (2x + y) + \hat{j} (x + y) + \hat{k} (3x + y) . \]
\[ \Rightarrow 2x + y = 1, x + y = 2, 3x + y = 3 . \]
By solving the first two equation, we get
\[ \Rightarrow x = - 1, y = 3 .\]
Clearly these values of x and y does not satisfy the third equation.
Hence the given vectors are non-coplanar.