Question

Geometria analítica

Verifique se os pontos A(-2,-5) B(0,-1) C(3,5) estão alinhados.

Answer 1

Os pontos estão alinhados se e somente se o determinante da seguinte matriz for igual a 0.

$\left[\begin{array}{ccc}x_{0}&y_{0}&1\\x_{1}&y_{1}&1\\x_{2}&y_{2}&1\end{array}\right] = 0$

Preenchendo ela com os nossos pontos e calculando seu determinante pelo teorema de Laplace:

$\left[\begin{array}{ccc}-2&-5&1\\0&-1&1\\3&5&1\end{array}\right] = 0\\0 * (-1)^{i+j} * \left[\begin{array}{cc}-5&1\\5&1\end{array}\right] + (-1)(-1)^{i+j} * \left[\begin{array}{cc}-2&1\\3&1\end{array}\right] + (1)(-1)^{i+j}*\left[\begin{array}{cc}-2&-5\\3&5\end{array}\right] \\(-1)^{i+j+1} * \left[\begin{array}{cc}-2&1\\3&1\end{array}\right] + (-1)^{i+j}*\left[\begin{array}{cc}-2&-5\\3&5\end{array}\right]$

$(-1)^{2+2+1} * \left[\begin{array}{cc}-2&1\\3&1\end{array}\right] + (-1)^{2+3}*\left[\begin{array}{cc}-2&-5\\3&5\end{array}\right] \\(-1)^{5} * \left[\begin{array}{cc}-2&1\\3&1\end{array}\right] + (-1)^{5}*\left[\begin{array}{cc}-2&-5\\3&5\end{array}\right] \\(-1) * \left[\begin{array}{cc}-2&1\\3&1\end{array}\right] + (-1)*\left[\begin{array}{cc}-2&-5\\3&5\end{array}\right] \\(-1) * (-2 -3) + (-1)* (-10 - (-15))\\(-1) * (-5) + (-1)* (-10+ 15)\\5 + (-1)* (5)\\5 - 5 = 0$

Portanto, estão alinhados.