Question

determine a equação cartesiana do plano (pi) que contém os pontos:
A: (1,1,2)
B: (3,2,2,)
C: (4,5,3)

Answer 1

P(x,y,z)

$\mathsf{\overrightarrow{AB}=B-A=(2,1,0)}\\\mathsf{\overrightarrow{AC}=C-A=(3,4,1)}\\\mathsf{\overrightarrow{AP}=P-A=(x-1,y-1,z-2)}$

$\mathsf{\lbrack\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AP}\rbrack}=\begin{bmatrix}2&1&0\\3&4&1\\x-1&y-1&z-2\end{bmatrix}=0$

$\mathsf{2([4z-8]-1[y-1])-1(3[z-2]-1[x-1])=0}\\\mathsf{(2(4z-8-y+1)-1(3z-6-x+1)=0}\\\mathsf{8z-16-2y+2-3z+6+x-1=0}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{\pi:~x-2y+5z-9=0}}}}}$