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I need to find the equation of the line that intersects perpendicularly the line: $$ \frac{x+1}{2}=-y=\frac{z-2}{3} $$ and passes through : $(2,3,1)$.

So I know that the line should be of the form: $$ (2,3,1)+t (\ell , m ,n) $$ where: $(\ell , m ,n) \cdot (2,-1,3)=0 $ but this gives me only one constraint on my parameters . How can I find all three of them?

Thanks in advance !

CrazyStatistician
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    You have not used the fact that the lines intersect. – najayaz Nov 17 '15 at 17:37
  • Let $\vec{ON}$ be the foot of perpendicular of $\vec{OA}=(2,3,1)$ to the line. ($\vec{ON}$ lies on the known line). $\vec{AN} \cdot (2,-1,3)=0$. – Nicholas Nov 17 '15 at 17:41
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    egreg: I can imagine only one such line. @Nicholas : I am sorry, but I can't see what you have just written . Are you trying to show me an arbitrary way to choose the vector $(\ell,m,n)$ ? Thanks! – CrazyStatistician Nov 17 '15 at 17:53

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