Further math vecotr question

It's going from
r=(3, 1,1)+a(1,3,-1)+b(2,5,0)
to r. (ci,d j, ek) = f

What do you understand about the different forms?

I know that (ci, d j, ek) is the normal to the plane
And that r.n=a.n where r is general and a is a specific point on plane (I'm assuming that is the (3,1,1))
But I don't know how to find the normal from it

For the first form, it says that any point on the plane can be formed from a linear combination of the two direction vectors and adding that to the point you mention. So how are the two direction vectors and the plane normal related? A sketch may help (as usual).

are they directions along the plan and therefore the dot product of...
Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

are they directions along the plan and therefore the dot product of...
Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

Its also worth noting that even if you didnt understand the geometry insight, could sub equation 1 into equation 2. Then when you take the dot product with each termm, the left hand side would depend on a and b whereas the right hand side wouldnt. The only way this could occur for all a and b (free parameters) would be if the dot product of the corresponding direction vectors with the normal is zero in both cases.

While it falls out of the algebra, its "better" to understand the properties (geometry).

Yes I tried that where the normal is 0 but I don't entirely know what that means
Cause every plane should have a normal right?