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

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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.

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