I have been taught that there is a clear distinction between the arithmetic involving objects in 2-D and objects in 3-D. As time went by I became better at solving 3-D geometry problems. For example, let us take a problem:
The image of a point A(2,3) on a line L: 3x + 4y = 12 is?
This would involve taking a point (a,b) which is the point of the image, and since the image and the object are equidistant from the line, we substitute (a+2/2, b+3/2) to the equation of line L. Now we go through a computation of finding the two slopes and making sure they are 90 degrees.
However after learning 3-D geometry, I could call the line:
$$\frac{3x-12}{4} = \frac{-y}{1}$$
and hence my line can be parameterized as $$L1 = \{\frac{4t+12}{3}, -t\}$$
Now I can use dot products and go ahead and solve how I would if it was a 3-D problem.
Now my question is, how and when can I change my 2-D problems to a 3-D geometry or even a vector problem (I have also been taught about the interconversion of 3-D cartesian equations to vector equations)?