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, \(2\vec a-\vec b+3\vec c,\) \(\vec a+\vec b-2\vec c\) and \(\vec a+\vec b-3\vec c\)
We can form a relation using these three vectors. Say,
Compare the vectors \(\vec a,\vec b\) and \(\vec c\),. We get
2 = x + y …(1)
–1 = x + y …(2)
3 = –2x – 3y …(3)
Solving equations (1) and (2) for x and y.
Equation (1), x + y = 2
Equation (2), x + y = –1
We get
The value of x and y cannot be found so it won’t satisfy equation (3).
Thus, \(2\vec a-\vec b+3\vec c,\) \(\vec a+\vec b-2\vec c\) and \(\vec a+\vec b-3\vec c\) are not coplanar.