Prove that the following vectors are non-coplanar : and

Question

Philoid · Accepted Answer

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

3 = 2x + 7y …(1)

1 = –x – y …(2)

–1 = 7x + 23y …(3)

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

Equation (1), 2x + 7y = 3

Equation (2), –x – y = 1

Multiply equation (2) by 2.

–x – y = 1 [× 2

⇒ –2x – 2y = 2 …(4)

Solving equations (4) and (1), we get

⇒ 5y = 5

⇒ y = 1

Put y = 1 in equation (2), we get

1 = –x – y

⇒ 1 = –x – (1)

⇒ 1 = –x – 1

⇒ x = –1 – 1

⇒ x = –2

Substituting x = –2 and y = 1 in equation (3), we get

–1 = 7x + 23y

Or 7x + 23y = –1

⇒ 7(–2) + 23(1) = –1

⇒ –14 + 23 = –1

⇒ 9 ≠ –1

∵, L.H.S ≠ R.H.S

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

Thus, , and are not coplanar.