Question

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

$(4,-3),(6,-4),(2,-2)$

Short answer

The points $(4,-3),(6,-4),(2,-2)$ are collinear.

Step-by-step solution

Step 1. Given

The points $(4,-3),(6,-4),(2,-2)$

$(x_1,y_1)=(4,-3)$

$(x_2,y_2)=(6,-4)$

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

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 $(x_1,y_1)=(4,-3)$,

$(x_2,y_2)=(6,-4)$

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

in the test for collinearity.

$\left|\begin{array}{ccc}4& -3& 1\\ 6& -4& 1\\ 2& -2& 1\end{array}\right|=4(-4+2)-(-3)(6-2)+1(-12+8)$

$=4(-2)+3\left(4\right)+1(-4)$

$=12-12$

$=0$

The value of determinant is zero.

So the points are collinear.