Question

Determine whether the points (3, −2), (5, −3), and (1, −1) are collinear.

Short answer

The Given points are collinear.

Step-by-step solution

Step 1. Given information

The given points are (3, −2), (5, −3), and (1, −1) .

Step 2. Calculation

$$\begin{gathered} \left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right] \\ \mathrm{Substitute}\mathrm{the}\mathrm{values}\mathrm{into}\mathrm{the}\mathrm{determinant}. \\ (3,-2),(5,-3),\mathrm{and}(1,-1). \\ D=\left[\begin{array}{ccc}3& -2& 1\\ 5& -3& 1\\ 1& -1& 1\end{array}\right] \\ \mathrm{Evaluate}\mathrm{the}\mathrm{determinant}\mathrm{by}\mathrm{expanding}\mathrm{by}\mathrm{minors}\mathrm{using}\mathrm{column}3. \\ 1\left[\begin{array}{cc}5& -3\\ 1& -1\end{array}\right]-1\left[\begin{array}{cc}3& -2\\ 1& -1\end{array}\right]+1\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right] \\ =(-5+3)-(-3+2)+(-9+10) \\ =-2+1+1 \\ =0 \\ \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{determinate}\mathrm{is}0,\mathrm{so}\mathrm{the}\mathrm{points}\mathrm{are}\mathrm{collinear}. \end{gathered}$$