Some confusion about where vectors emanate from, Take the plane x+y+z=0, for example.  Then the vectors (0,0,0), (1,-1,0), (1,0,-1) "lie on this plane."  And to find a normal to this plane, just compute the cross-product of any two independent vectors from this plane.
But are the vectors "on this plane" emanating from the origin?  I'm pretty sure it is not.  And, the normal vector, (1,1,1), emanates from the plane, not from the origin.  Or, a unit normal for a sphere, sits on the sphere, and doesn't start from the origin.
So my question is ...what do the numbers in these vectors actually mean?  What would (1,-1,0) mean?  Is it still 1 unit in the x direction, -1 unit in the y direction, and 0 units in the z direction?
Thanks,

When using vectors to describe position, as in your three vectors, they describe the position obtained by moving from origin by the amount and direction given by the vector. Describing the normal vector to a plane is just the usual notion of vector as a motion with direction and length.
