Question

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

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

Short answer

The points $(0,1),(2,0),(-2,2)$ are collinear.

Step-by-step solution

Step 1. Given

The points $(0,1),(2,0),(-2,2)$

$(x_1,y_1)=(0,1)$

$(x_2,y_2)=(2,0)$

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

To find if the points are collinear.

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

Substitute the points, $(x_1,y_1)=(0,1)$,

$(x_2,y_2)=(2,0)$

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

in the test for collinearity.

$\left|\begin{array}{ccc}0& 1& 1\\ 2& 0& 1\\ -2& 2& 1\end{array}\right|=0(0-2)-1(2+2)+1(4-0)$

$=0-1\left(4\right)+1\left(4\right)$

$=-4+4$

$=0$

The value of determinant is zero.

So the points are collinear.