Question

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.

Answer 1

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.

In a parallelogram ABCD, the diagonals bisect each other. Hence,

\( \vec{A} + \vec{C} = \vec{B} + \vec{D} \)

Therefore, \( \vec{D} = \vec{A} + \vec{C} - \vec{B} \)

Substitute the given coordinates:

\( \vec{D} = (3, -1, 2) + (-1, 1, 2) - (1, 2, -4) \)

\( = (2, 0, 4) - (1, 2, -4) \)

\( = (2-1,\; 0-2,\; 4-(-4)) \)

\( = (1, -2, 8) \)

Final Answer:
The coordinates of the fourth vertex are D(1, –2, 8).