Question

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.

Answer 1

$A(5,-1,1),B(7,-4,7),C(1,-8,10),C(1,-8,10),D(-1,-3,4)$

$\overrightarrow{AB}=2\hat{i}-3\hat{j}+6\hat{k}$

$\overrightarrow{AD}=-6\hat{i}-2\hat{j}+3\hat{k}$

$\overrightarrow{AB}.\overrightarrow{AD}=\left(2\right)(-6)+(-3)(-2)+\left(6\right)\left(3\right)=-12+6+18=12$

$|AB|=\sqrt{(7-5{)}^2+(-4+1{)}^2+(7-1{)}^2}=\sqrt{2^2+3^2+6^2}=7$

$|BC|=\sqrt{(7-1{)}^2+(-4+8{)}^2+(7-10{)}^2}=\sqrt{6^2+4^2+3^2}=\sqrt{61}$

$|CD|=\sqrt{(1+1{)}^2+(-8+3{)}^2+(10-4{)}^2}=\sqrt{2^2+5^2+6^2}=\sqrt{65}$

$\therefore$ Neither Square nor Rhombus.