Question

If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[2 \vec{a} - \vec{b} + 3 \vec{c} , \vec{a} + \vec{b} - 2 \vec{c}\text{ and }\vec{a} + \vec{b} - 3 \vec{c}\]

Answer 1

Let if possible the following vectors are coplanar. Then one of the vector is expressible in terms of the other two.
We have,

\[2 \vec{a} - \vec{b} + 3 \vec{c} = x( \vec{a} + \vec{b} - 2 \vec{c} ) y( \vec{a} + \vec{b} - 3 \vec{c} ) . \]
\[ = \vec{a} (x + y) + \vec{b} (x + y) + \vec{c} ( - 2x - 3y) . \]
\[ \Rightarrow x + y = 2 , x + y = - 1 , - 2x - 3y = 3 .\]
 which is not true, as \[x + y = 2\] ≠ - 1.
Hence the given vectors are non-coplanar.