Problems from Implicit equations of a straight line in the space

Development:

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: $${\displaystyle \frac{x-2}{-1}}={\displaystyle \frac{y-1}{-3}}={\displaystyle \frac{z+2}{5}}$$

Finally, if we separate the continuous equations and simplify a little bit we have: $${\displaystyle \frac{x-2}{-1}}={\displaystyle \frac{y-1}{-3}}\Rightarrow -3x+6=-y+1\Rightarrow -3x+y+5=0$$ $${\displaystyle \frac{x-2}{-1}}={\displaystyle \frac{z+2}{5}}\Rightarrow 5x-10=-z-2\Rightarrow 5x+z-8=0$$ Therefore the implicit equations are: $$-3x+y+5=0$$ $$5x+z-8=0$$

Solution:

$-3x+y+5=0$; $5x+z-8=0$

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