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If I start with a concrete parametric form of the equation of a line in 3D

$$\vec x= \vec p + t \vec d$$

What sequence of steps takes me to the line's general form $$a_1x + b_1y + c_1z = d_1$$ $$a_2x + b_2y + c_2z = d_2$$

Specifically, what conditions are placed on the normal vectors $\vec n_1=\pmatrix{a_1\\b_1\\c_1}$, $\vec n_2=\pmatrix{a_2\\b_2\\c_2}$? Do they have to be orthogonal to each other or are they just any vectors which are orthogonal to the direction vector $\vec d$?

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The equations you are asking for are not uniquely determined. You can use any two planes which are not parallel and contain the given line to get an equation of that line by taking the intersection of the planes, which just amounts to asking that both equations are simultaneously fulfilled.

As you already noted, one way to get there is to determine two (non-parallel) normal vectors, $n_1, n_2$, say, to the given line and then write down the equations for the planes normal to these vectors and passing through a given point of the line (e.g., $p$). You can use the Hesse normal form for this. If, for example, $n_1 = (v_1, v_2, v_3),$ $||n_1||=1$ and $\langle p,n_1 \rangle = d_1$ is the signed distance from the position vector $p$ to the origin projected onto $n_1$, then the first plane is given by $$ v_1 x + v_2 y + v_3 z = d_1$$ similarly for the second normal, giving you the second equation. This will work as long as you take two linear independent normals to the given line. It does not matter wich one.

There are other ways to get to such equations, as well, it's just geometry.

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