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

Let the fourth vertex be (x, y, z). We know that diagonals of a parallelogram are bisecting to each other. i.e. mid-point of a diagonals are coinciding.  Mid-point of diagonal AC = Mid-point of diagonal BD ∴ 1+22,2+32,3+22=−1+x2,−2+y2,−1+z2⇒ 32,52,52=−1+x2,−2+y2,−1+z2⇒ 32=−1+x2,52=−2+y2,52=−1+z2⇒ x=4,y=7,z=6Hence, required point is (4, 7, 6)