Determine if the points $(5, - 1, 1), (7, - 4, 7), (1, - 8, 10)$ and $(- 1, - 3, 4)$ are the vertices of a rhombus or a square.

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Solution

$A (5, - 1, 1), B (7, - 4, 7), C (1, - 8, 10), C (1, - 8, 10), D (- 1, - 3, 4)$
$\to A B = 2^i - 3^j + 6^k$
$\to A D = - 6^i - 2^j + 3^k$
$\to A B . \to A D = (2) (- 6) + (- 3) (- 2) + (6) (3) = - 12 + 6 + 18 = 12$
$| A B | = \sqrt{(7 - 5)^{2} + (- 4 + 1)^{2} + (7 - 1)^{2}} = \sqrt{2^{2} + 3^{2} + 6^{2}} = 7$
$| B C | = \sqrt{(7 - 1)^{2} + (- 4 + 8)^{2} + (7 - 10)^{2}} = \sqrt{6^{2} + 4^{2} + 3^{2}} = \sqrt{61}$
$| C D | = \sqrt{(1 + 1)^{2} + (- 8 + 3)^{2} + (10 - 4)^{2}} = \sqrt{2^{2} + 5^{2} + 6^{2}} = \sqrt{65}$
$∴$ Neither Square nor Rhombus.

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