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If a,b and c are non-coplanar vectors, p...

If a,b and c are non-coplanar vectors, prove that 3a-7b-4c, 3a-2b+c and a+b+2c are coplanar.

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Let `alpha=3a-7b-4c,beta=3a-2b+c`
and `gamma=a+b+2c`
also, let `alpha=x beta, y-gamma`
`implies3a-7b-4c=x(3a-2b+c)+y(a+b+2c)`
`=(3x+y)a+(-2x+y)b+(x+2y)c`
since, a,b and c are non-coplanar vectors.
therefore, `3x+y=3,-2x+y=-7`
and `x+2y=-4`
solving first two, we find that x=2 and y=-3. these values of x annd y satisfy the third equation as well.
so, x+2 and y=-3 is the unique solution for the above system of equation.
`implies alpha=2beta-3gamma`
Hence, the vectors `alpha,beta and gamma` are complanar, because `alpha` is uniquely written as linear combination of other two.
Trick for the vector `alpha,beta, gamma` to be coplanar, we must have
`|(3,-7,-4),(3,-2,1),(1,1,2)|=0`, which is true
Hence, `alpha,beta,gamma` are coplanar.
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