Implicit equations of a straight line in the space
Consider the points $A = (2, 1,-2)$ and $B = (1,-2, 3)$, and find the implicit equations of the straight line that goes through $A$ anb $B$.
We will start computing a director vector of the straight line: $$\overrightarrow{AB}=B-A=(1,-2,3)-(2,1,-2)=(-1,-3,5)$$
Therefore, with the director vector and point $A$, we obtain the continuous equation: $$\dfrac{x-2}{-1}=\dfrac{y-1}{-3}=\dfrac{z+2}{5}$$
Finally, if we separate the continuous equations and simplify a little bit we have: $$\dfrac{x-2}{-1}=\dfrac{y-1}{-3} \Rightarrow -3x+6=-y+1 \Rightarrow -3x+y+5=0$$ $$\dfrac{x-2}{-1}=\dfrac{z+2}{5} \Rightarrow 5x-10=-z-2 \Rightarrow 5x+z-8=0$$ Therefore the implicit equations are: $$-3x+y+5=0$$ $$5x+z-8=0$$
$-3x+y+5=0$; $5x+z-8=0$