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Three vertices of a parallelogram ABCD a...

Three vertices of a parallelogram ABCD are A (3,-1,2), B (1, 2, 4) and C(-1,1,2). Find the coordinates of the fourth vertex D.

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Let coordinates of fourth vertex D be (x,y,z).
`because ABCD` is a parallelogram.
`therefore` Coordinates of mid-point of diagonal BD
=Coordinates of mid-point of diagonal AC
`implies ((1+x)/(2),(2+y)/(2),(-4+z)/(2))`
`=((3-1)/(2),(-1+1)/(2),(2+2)/(2))`
`implies (1+x)/(2)=(3-1)/(2),(2+y)/(2)=(-1+1)/(2)`
and `(-4+z)/(2)=(2+2)/(2)`
`implies 1+x=2, 2+y=0 " " "and" -4+z=4`
`implies x=1, y= -2 " " "and" z=8`
`therefore` Coordinates of D=(1,-2,8)
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