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Question
Show that four points A, B, C and D whose position vectors are
`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.
Solution
The position vectors of the points A, B, C and D are `4hati+5hatj+hatk,-hatj-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` , respectively. Then
`vec(BA)=(4hati+5hatj+hatk)-(0hati-hati-hatk)=4hati+6hatj+2hatk`
`vec(BC)=(3hati+9hatj+4atk)-(0hati-hati-hatk)=3hati+10hatj+5hatk`
`vec(BD)=(-4hati+4hatj+4hatk)-(0hati-hati-hatk)=-4hati+5hatj+5hatk`
The given points are coplanar iff vectors `vec(BA),vec(BC), vec(BD)` are coplanar
Now,
`[[vec(BA),vec(BC), vec(BD)]]`
`=|[4,6,2],[3,10,5],[-4,5,5]|`
=4(50-25)-6(15+20)+2(15+40)
=100-210+110
=0
Hence, the four points A, B, C and D are coplanar.
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