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If veca, vecb and vecc are three non-zer...

If `veca, vecb and vecc` are three non-zero, non-coplanar vectors,then find the linear relation between the following four vectors : `veca-2vecb+3vecc, 2veca-3vecb+4vecc, 3veca-4vecb+ 5vecc, 7veca-11vecb+15vecc`.

Text Solution

Verified by Experts

Any vector `vecr` can be uniquely expressed as a linear combinatioin of three non-coplanar vectors.
Let us choose that
`" "7veca -11 vecb + 15 vecc= x(veca-2vecb+ 3vecc) + y (2veca-3vecb+4vecc)+ z(3veca-4vecb+5vecc)`
Comparing the coefficients of `veca, vecb and vecc` on both sides, we get
`x+2y+3z=7, -2x-3y-4z=-11, 3x+4y+ 5z=15`
Eliminating x and then solving for `y and z`, we get x=1, y=3, z=0`
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