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प्रश्न
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.
उत्तर
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`
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