Let ${\mathrm{P}}_{\mathrm{r}}\left({\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1\right),\mathrm{r}=1,2,3$ be three points where ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3;{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3;{\mathrm{z}}_1,{\mathrm{z}}_2,{\mathrm{z}}_3$ are each in G.P. with the same common ratio then ${\mathrm{P}}_1,{\mathrm{P}}_2,{\mathrm{P}}_3$ are

Suppose that the points are $P_1\left(a,b,c\right)$, $P_2\left(ap,bp,cp\right)$ and $P_3\left(a{p}^2,b{p}^2,c{p}^2\right)$

To check the given three points are collinear or not, find the determinant of the matrix formed by the coordinates

$\begin{array}{rcl}\left|\begin{array}{ccc}a& b& c\\ ap& bp& cp\\ a{p}^2& b{p}^2& cp\end{array}\right|& =& p^3\left|\begin{array}{ccc}a& b& c\\ a& b& c\\ a& b& c\end{array}\right|\\ & =& p^3\left(0\right)\\ & =& 0\end{array}$

It means the given points are to be collinear