Question

Determine if the points $(2,3),(-1,0)$ and $(2,-4)$ are collinear.

Answer 1

We know that three point $A\left(X_1{Y}_1\right)B\left(X_2{Y}_2\right)$ and $C\left(X_3{Y}_3\right)$ are collinear if

$\Rightarrow \left|\begin{array}{ccc}{X}_1& Y_1& 1\\ X_2& Y_2& 1\\ X_3& Y_3& 1\end{array}\right|=0$

for the points $(2,3),(-1,0)$ and $(2,-4),$ the value of the above determinant is

$\left|\begin{array}{ccc}2& 3& 1\\ -1& 0& 1\\ 2& -4& 1\end{array}\right|$

expanding along the third column :

$=\left|\begin{array}{cc}-1& 0\\ 2& -4\end{array}\right|-1\left|\begin{array}{cc}2& 3\\ 2& -4\end{array}\right|+1\left|\begin{array}{cc}2& 3\\ -1& 0\end{array}\right|$

$=4-(-8-6)+\left(3\right)$

$=4+14+3$

$=21\ne 0$

Hence, the points $(2,3)(-1,0)$ and $(2,-4)$ are collinear.