Verify the following:\left(\right. - 1 , 2 , 1 \left.\right) , \left(\right. 1 , - 2 , 5 \left.\right) , \left(\right. 4 , - 7 , 8 \left.\right) and \left(\right. 2 , - 3 , 4 \left.\right) are vertices of a parallelogram.

Engineering

Mathematics

Vector Addition and Subtraction

theorem in space

Question

Verify the following:

(- 1, 2, 1), (1, - 2, 5), (4, - 7, 8)

and

(2, - 3, 4)

are vertices of a parallelogram.

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Let the vertices of parallelogram be,

A (- 1, 2, 1), B (1, - 2, 5), C (4, - 7, 8)

and

D (2, - 3, 4)

So,

\begin{array}{l} A B = \sqrt{{(- 1 - 1)}^{2} + {(2 + 2)}^{2} + {(1 - 5)}^{2}}, = \sqrt{4 + 16 + 16}, = 6 \\ B C = \sqrt{{(1 - 4)}^{2} + {(- 2 + 7)}^{2} + {(5 - 8)}^{2}}, = \sqrt{9 + 25 + 9}, = \sqrt{53} \\ C D = \sqrt{{(4 - 2)}^{2} + {(- 7 + 3)}^{2} + {(8 - 4)}^{2}}, = \sqrt{4 + 16 + 16}, = 6 \\ D A = \sqrt{{(2 + 1)}^{2} + {(- 3 - 2)}^{2} + {(4 - 1)}^{2}}, = \sqrt{9 + 25 + 9}, = \sqrt{53} \end{array}

From this we can observe that the opposite sides of the quadrilateral are equal. Hence,

A B C D

is a parallelogram.