Question

In the following exercises, determine whether the given points are collinear.

$(0,2),(-1,-1),(-2,4)$

Short answer

The points $(0,2),(-1,-1),(-2,4)$ are not collinear.

Step-by-step solution

Step 1. Given Information

The given points are $(0,2),(-1,-1),(-2,4)$

$(x_1,y_1)=(0,2)$

$(x_2,y_2)=(-1,-1)$

$(x_3,y_3)=(-2,4)$

Step 2. Test for collinear points

Three points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$are collinear if and only if

$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Step 3. Substitute the points in the formula

Substitute the points $(x_1,y_1)=(0,2)$, $(x_2,y_2)=(-1,-1)$$(x_3,y_3)=(-2,4)$in the formula.

$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\left|\begin{array}{ccc}0& 2& 1\\ -1& -1& 1\\ -2& 4& 1\end{array}\right|$

$=0(-1-4)-2(-1+2)+1(-4-2)$

$=0-2\left(1\right)+1(-6)$

$=-2-6$

$=-8$

The value of the determinant is non zero number.

So, the points are not collinear.