Question

4. Verificați dacă punctele A(1; 1), B(2; -1) şi C(3; -3) sunt coliniamre

Answer 1

$\displaystyle A(x_A;y_A),~B(x_B;y_B),~C(x_C;y_C)\\\\Punctele~A,~B,~C~sunt~coliniare~\Leftrightarrow~\left|\begin{array}{ccc}x_A&y_A&1\\x_B&y_B&1\\x_C&y_C&1\end{array}\right|=0\\\\ A(1;1),~B(2;-1),~C(3;-3)\\\\ \left|\begin{array}{ccc}1&1&1\\2&-1&1\\3&-3&1\end{array}\right|=1 \cdot (-1)\cdot 1+1\cdot2\cdot (-3)+1\cdot 1\cdot3-1\cdot(-1)\cdot3-\\\\~~~~~~~~~~~~~~~~~~~~-1\cdot2\cdot1-1 \cdot 1 \cdot (-3)=-1-6+3+3-2+3=0 \Rightarrow\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow Punctele~A,~B,~C~sunt~coliniare.$