Why can't a set of vectors span R^3 if the vectors are in the same plane? I'm really struggling to understand the concept of spanning and my book doesn't go into any detail about why S doesn't span R^3 in this case. It says a set of vectors in the same plane only span the plane they're in. But, for example, take the plane z = 1. A set of vectors such that all are in the plane could be S = {(0, 0, 1), (1, 1, 1), (2, 2, 1)}. But the linear combination 1(0, 0, 1) + 1(1, 1, 1) + 1(2, 2, 1) = (3, 3, 3), for example, is not in the plane z = 1. So why is it true that S spans only the plane containing its vectors? Why can't S span R^3?
 A: Try to visualize-what does it mean for a set of vectors to span a plane? It means they are contained entirely within the plane, something that can only happen if the plane is going through the origin. You can also understand this analytically-try to show that any three vectors in the same plane through the origin are linearly dependent. This is how I understood it first.
A: the vectors u wrote aren't in the plane z=1. 
a vector in a plane means all points on that vector (basically a line) lie in that plane. 
for a vector to lie on this plane its z component will be 0.
A: The span of a set of vectors is defined as all linear combinations of that set of vectors. In particular, if $x\in V$, where $V$ is the span of a set of vectors, then $cx\in V$ for every real number $c$. This means that the set you chose is not the span of a set of vectors (whenever $c\neq1$, for any vector $x$ in the set with $z=1$, $cx$ has $c$ in the $z$-component). If you try the set of vectors with $z=0$, (which is the span of, say, $\{(1,0,0),(0,1,0)\}$), you'll see that this is indeed just the plane containing those two vectors. As some others have mentioned, the span of a set of vectors necessarily contains the $\vec{0}$ vector, since $0x=\vec{0}$ for any vector $x$.
In order to interpret the statement in your textbook correctly, you actually need to specify that the vectors are in a plane containing the origin, since otherwise, the set $\{(1,0,1),(0,1,1),(0,0,1)\}$ is contained in the plane of vectors with $z=1$ (which does not contain the origin), but $(x,y,z)=x(1,0,1)+y(0,1,1)+(z-x-y)(0,0,1),$ so these vectors do span $\mathbb{R}^{3}$.
A: You probably got confused because you always see that $R^3$ is spanned by $e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1)$ but remember that they stand for $(1,0,0) - O$,$(0,1,0)-O$, $(0,0,1)-O$, where $O$ is the origin and when you take the difference between points $B$ and $A$ you get the vector $\bar{AB}$.
