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

Question

If \(\vec a, \vec b,\vec c\) are non–zero, non-coplanar vectors, prove that the following vectors are coplanar : \(\vec a-2\vec b+3\vec c,\) \(\vec a-3\vec b+5\vec c\) and \(-2\vec a+3\vec b-4\vec c\)

Answer 1

We have been given that, \(\vec a-2\vec b+3\vec c,\) \(\vec a-3\vec b+5\vec c\) and \(-2\vec a+3\vec b-4\vec c\)

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

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

We get

1 = – 2y …(1

 –2 = –3x + 3y …(2)

3 = 5x – 4y …(3)

Solving equation (1) for y,

Equation (1), –2y = 1

⇒ y = \(-\cfrac12\)

Put y = \(-\cfrac12\) in equation (2),

We get

⇒ –6x – 3 = –2 × 2

⇒ –6x – 3 = –4

⇒ –6x = –4 + 3

⇒ –6x = –1

⇒ x = \(\cfrac16\)

Substituting x = \(\cfrac16\) and y = \(-\cfrac12\) in equation (3),

We get

3 = 5x – 4y

Or 5x – 4y = 3

∵, L.H.S ≠ R.H.S

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

Thus,   \(\vec a-2\vec b+3\vec c,\) \(\vec a-3\vec b+5\vec c\) and \(-2\vec a+3\vec b-4\vec c\) are not coplanar.