Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex.

The three
vertices of a parallelogram ABCD are given as A (3, –1, 2), B
(1, 2, –4), and C (–1, 1, 2). Let the coordinates of the
fourth vertex be D (*x*, *y*, *z*).

We know that the diagonals of a parallelogram bisect each other.

Therefore, in parallelogram ABCD, AC and BD bisect each other.

∴Mid-point of AC = Mid-point of BD

⇒ *x*
= 1, *y* = –2, and *z* = 8

Thus, the coordinates of the fourth vertex are (1, –2, 8).

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