Prove that the following vectors are non–coplanar :

and

We have been given that, , and .

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

Comparing coefficients of , and , we get

1 = 2x + y …(1)

2 = x + y …(2)

3 = 3x + y …(3)

Solving equations (1) and (2) for x and y.

Equation (1), 2x + y = 1

Equation (2), x + y = 2

⇒ x = –1

Put x = –1 in equation (2), we get

2 = x + y

⇒ 2 = (–1) + y

⇒ y = 2 + 1

⇒ y = 3

Substituting x = –1 and y = 3 in equation (3), we get

3 = 3x + y

Or 3x + y = 3

⇒ 3(–1) + (3) = 3

⇒ –3 + 3 = 3

⇒ 0 ≠ 3

∵, L.H.S ≠ R.H.S

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

Thus, , and are not coplanar.