If vector a, vector b, vector c are non–coplanar vectors, prove that the following vectors are non-coplanar: vector a+ 2vector b+3 vector c

Question

If \(\vec a,\vec b,\vec c\) are non–coplanar vectors, prove that the following vectors are non-coplanar: \(\vec a+2\vec b+3\vec c,\) \(2\vec a+\vec b+3\vec c\) and \(\vec a+\vec b +\vec c\)

Answer 1

We have been given that,   \(\vec a+2\vec b+3\vec c,\) \(2\vec a+\vec b+3\vec c\) and \(\vec a+\vec b +\vec c\)

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

Compare the vectors \(\vec a,\vec b\) and \(\vec c\) We get

1 = 2x + y …(1)

2 = x + y …(2)

3 = 3x + y …(3)

Solving equation (1) and (2) for x and y,

⇒ 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,     \(\vec a+2\vec b+3\vec c,\) \(2\vec a+\vec b+3\vec c\) and \(\vec a+\vec b +\vec c\) are not coplanar.