Three vertices of a parallelogram ABCD are A(3, 1, 2), B(0, 1, 5) and C(-1, 2, -3) Find the
coordinates of the fourth vertex.

Explanation

Let the coordinates of the fourth vertex D be (x, y, z)
Since diagonals of a parallelogram bisect each other. Therefore, mid-point of AC and BD coincide.
`therefore ((3-1)/2,(1+2)/2,(2-3)/2) = ((0+x)/2, (y+1)/2,(z+5)/2)`
`rArr (1, 3/2, (-1)/2)= (x/2,(y+1)/2,(z+5)/2)`
`rArr x/2 = 1, (y+1)/2 = 3/2, (z+5)/2 = (_1)/2`
`rArr x= 2, y=2 and z= - 6`
Hence the coordinates of the fouth vertex are (2, 2, -6).