vertical vectors having horizontal components? What if we have a vector which is, say, vertically oriented. Now if we find the projection of this vector on a line 30 degrees from the vertical, we will get another vector at an angle 60 degrees to the horizontal. Now we can find the projection of this vector on the x-axis which will be a non-zero quantity. But this is absurd since a vertical vector cannot have a horizontal component. So where is the mistake I am making in my line of thought.
 A: I think that your problem is strictly related with the following aspect:
When you find the projection of a vector along another one you loose information about the initial one; in mathematical terms, if you see the projection as an operator that acts on a vector of a vector space giving you its projection (think it as a function if it's the first time you hear the word operator) then the operator is not invertible (continuing the parallelism, the function is not bijective). 
Qualitatively speaking, coming back to your example, if you consider only the projection (forget for one moment the vertical vector), would you be able -with no further informations- to identify uniquely a vector whose projection is the one you're looking at? No, since there are actually infinite vectors which projected along that axis have that same projection.
In conclusion, you can't study the properties of one vector by looking at its projection along another vector since the least doesn't contain all the information about the previous one!
