Prove that the following vectors are coplanar :

and

Vectors parallel to the same plane, or lie on the same plane are called coplanar vectors

The three vectors are coplanar if one of them is expressible as a linear combination of the other two.

We have been given that, , and .

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

Comparing coefficients of , and , we get

2 = x + 3y …(1)

–1 = –3x – 4y …(2)

1 = –5x – 4y …(3)

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

Equation (1), x + 3y = 2

Equation (2), –3x – 4y = –1

Multiply equation (1) by 3.

x + 3y = 2 [× 3

⇒ 3x + 9y = 6 …(4)

Solving equations (4) and (2), we get

⇒ 5y = 5

⇒ y = 1

Put in equation (1), we get

2 = x + 3y

⇒ x + 3(1) = 2

⇒ x = 2 – 3

⇒ x = –1

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

–5x – 4y = 1

⇒ –5(–1) – 4(1) = 1

⇒ 5 – 4 = 1

⇒ 1 = 1

∵, L.H.S = R.H.S

⇒ The value of x and y satisfy equation (3).

Thus, , and are coplanar.