If are non–zero, non-coplanar vectors, prove that the following vectors are coplanar :

and

We have been given that, , and .

We can form a relation using these three vectors. Say,

Compare the vectors , and . We get

1 = – 2y …(1)

–2 = –3x + 3y …(2)

3 = 5x – 4y …(3)

Solving equation (1) for y,

Equation (1), –2y = 1

Put in equation (2), we get

⇒ –6x – 3 = –2 × 2

⇒ –6x – 3 = –4

⇒ –6x = –4 + 3

⇒ –6x = –1

Substituting and in equation (3), we get

3 = 5x – 4y

Or 5x – 4y = 3

But

∵, L.H.S ≠ R.H.S

⇒ The value of x and y doesn’t satisfy equation (3).

Thus, , and are not coplanar.