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That’s to be expected since you can multiply the equation of a plane by any nonzero constant to get another equation for the same plane. Can i use the origin (0,0,0) as a point on the plane, or can i simply use one of the vectors as a point on the plane? There are many different measures of how well a plane fits given data, and different measures give rise to different best fitting planes

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So you had best tell us what you have in mind as your measure of how well a given plane fits some given data. I think my main issue here is, how do i determine a point on the plane 0 you do not have enough information to specify the exact value of d in your equation

I would leave the equation as$$ x+11y+3z+d=0$$ until a point on the plane is given

At this point we have a family of parallel planes. Can you please explain to me how to get from a nonparametric equation of a plane like this $$ x_1−2x_2+3x_3=6$$ to a parametric one The equation of a plane that goes through the origin can be written as $ax+by+cz=0$

Notice that the origin $ (0,0,0)$ satisfies this equation and hence belongs to the plane. Method 1 gives you the direction of a line in your original plane There is no reason why this direction should be orthogonal to the plane you require Method 2 will work though as you now have 2 directions in the plane you have to find.

Here's what i have so far:

I see that the related problem has a point on the plane supplied as well