Chapter 11.4, Problem 69E

Collinear Points and Determinants

(a) If three points lie on a line, what is the area of the “triangle” that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points $\left(a_1,b_1\right)$, $\left(a_2,b_2\right)$ and $\left(a_3,b_3\right)$ are collinear if and only if

$\left|\begin{array}{ccc}{a}_1& b_1& 1\\ a_2& b_2& 1\\ a_3& b_3& 1\end{array}\right|=0$

(b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer.

(i) $\left(-6,4\right)$, $\left(2,10\right)$, $\left(6,13\right)$

(ii) $\left(-5,10\right)$, $\left(2,6\right)$, $\left(15,-2\right)$