Question

Determine se os pontos (2, 1), B(3,1) e C(5,4) são colineares?

Answer 1

✅ Após resolver os cálculos, concluímos que o referidos pontos do plano cartesiano:

$\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf N\tilde{a}o\:s\tilde{a}o\:Colineares\:\:\:}}\end{gathered} }$

Verificando a colinearidade de três pontos no plano cartesiano.

Sejam os pontos dados:

              $\Large\begin{cases} A(2, 1)\\B(3, 1)\\C(5, 4)\end{cases}$

Dizemos que três pontos no plano cartesiano são colineares se, e somente se, o determinante da matriz quadrada formada pelas coordenadas destes pontos for igual a "0", isto é:

            $\Large\displaystyle\begin{gathered} \det M = 0\end{gathered}$

Se:

        $\Large\displaystyle\begin{gathered} M = \begin{bmatrix} 2 & 1 & 1\\3 & 1 & 1\\5 & 4 & 1\end{bmatrix}\end{gathered}$

Então, temos:

      $\Large\displaystyle\begin{gathered} \det M = \begin{vmatrix} 2 & 1 & 1\\3 & 1 & 1\\5 & 4 & 1\end{vmatrix}\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = \begin{vmatrix} 1 & 1\\4 & 1\end{vmatrix}\cdot2 - \begin{vmatrix} 3 & 1\\5 & 1\end{vmatrix}\cdot1 + \begin{vmatrix} 3 & 1\\5 & 4\end{vmatrix}\cdot 1\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = (1 - 4)\cdot2 - (3 - 5)\cdot1 + (12 - 5)\cdot1\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = (-3)\cdot2 - (-2)\cdot1 + 7\cdot1\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = -6 + 2 + 7\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = 3\end{gathered}$

Então, temos:

       $\Large\displaystyle\begin{gathered} \det M = 3\end{gathered}$

Se:

         $\Large\displaystyle\begin{gathered} \det M \neq 0\end{gathered}$

✅ Então, os pontos não são colineares.

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$