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The points A (4, 5, 10), B (2, 3, 4) and C (1, 2, –1) are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D.

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The points A(4, 5, 10), B(2, 3, 4) and C(1, 2, -1) are three vertices of parallelogram ABCD.

Let coordinates of D be(x, y, z)

Direction vector along AB is

\(\therefore\) Equation of line AB, is given by

Direction vector along BC is

\(\therefore\) Equation of a line BC, is given by

Since ABCD is a parallelogram AC and BD bisect each other

Coordinates of D are (3, 4, 5).

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