Question

Shown that the point $(5,-1,1),(7,-4,7),(1,-6,10)$ and $(-1,-3,4)$ are the vertices of a rhombus.

Answer 1

For a quadrant to be rhombus, four side of quadrant be equal.

By distance formula:

$S_1=\sqrt{{(7-5)}^2+{(-4+1)}^2{+(7-1)}^2}=7S_2=\sqrt{{(1-7)}^2{+(-6+4)}^2{+(10-7)}^2}=7S_3=\sqrt{{(-1-1)}^2{+(-6+4)}^2{+(10-7)}^2}=7S_4=\sqrt{{(-1-5)}^2{+(-3+1)}^2{+(4-1)}^2}=7\because S_1{=S}_2=S_3=S_4$

Hence the given quadrant is rhombus