Question

Determine whether the points (−4, −1), (−6, 2), and (−2, −4) are collinear.

Short answer

The Given points are collinear.

Step-by-step solution

Step 1. Given information

The given points are (−4, −1), (−6, 2), and (−2, −4) .

Step 2. Calculation

$$\begin{gathered} \left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right] \\ \mathrm{Substitute}\mathrm{the}\mathrm{values}\mathrm{into}\mathrm{the}\mathrm{determinant}. \\ (-4,-1),(-6,2),\mathrm{and}(-2,-4). \\ D=\left[\begin{array}{ccc}-4& -1& 1\\ -6& 2& 1\\ -2& -4& 1\end{array}\right] \\ \mathrm{Evaluate}\mathrm{the}\mathrm{determinant}\mathrm{by}\mathrm{expanding}\mathrm{by}\mathrm{minors}\mathrm{using}\mathrm{column}3. \\ 1\left[\begin{array}{cc}-6& 2\\ -2& -4\end{array}\right]-1\left[\begin{array}{cc}-4& -1\\ -2& -4\end{array}\right]+1\left[\begin{array}{cc}-4& -1\\ -6& 2\end{array}\right] \\ =(24+4)-(16-2)+(-8-6) \\ =28-14-14 \\ =0 \\ \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{determinate}\mathrm{is}0,\mathrm{so}\mathrm{the}\mathrm{points}\mathrm{are}\mathrm{collinear}. \end{gathered}$$