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Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

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#### Solution

Position vector of `vec(OA)=4hati+hatj+2hatk`

Position vector of `vec(OB)=5hati+xhatj+6hatk`

Position vector of `vec(OC)=5hati+hatj-hatk`

Position vector of `vec(OD)=7hati+4hatj+0hatk`

`vec(AB)=vec(OB)-vec(OA)`

`=5hati+xhatj+6hatk-4hati-hatj-2hatk=hati+(x-1)hatj+4hatk`

`vec(AC)=vec(OC)-vec(OA)`

`=5hati+hatj-hatk-4hati-hatj-2hatk=hati-3hatk`

`vec(AD)=vec(OD)-vec(OA)`

`=7hati+4hatj+0hatk-4hati-hatj-2hatk=3hati+3hatj-2hatk`

The above three vectors are coplanar

`=>vec(AB).(vec(AC)xxvec(AD))=0`

`=>|[1,x-1,4],[1,0,-3],[3,3,-2]|=0`

`=>1(0 + 9) - (x - 1)( -2 + 9) + 4(3 - 0) = 0`

`=>9-7(x-1)+12=0`

`=>-7(x-1)=-21`

`=>x-1=3`

`therefore x=4`

Concept: Vectors Examples and Solutions

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