What is left multiplication for vectors?? Let $G$ be a subgroup of $\mathbb{R}^3$ consisting of vectors of the form $(x,y,0)$.  Let $G$ act on $\mathbb{R}^3$ by left multiplication. Describe the orbits of this $G$ - action geometrically.  Show that the set of orbits is in one to one correspondence with $\mathbb{R}$.
My question: What is meant by left multiplication in this context? Since these are vectors we are talking about there are multiple ways of multiplying vectors.  Am i supposed to assume that it is the dot product but then the dimensions don't line up unless I take the transpose of the second vector.  Interpreting what this question is asking me is my biggest problem so if someone could clarify what this question means I would appreciate it, thanks in advance!
 A: If you are talking about vectors then the natural operation is addition.
It's a little misleading to say "left multiplication". What they mean is to operate on the left. But in the case of addition, which is commutative, operating on the left is the same as on the right.
You have $\alpha : G \times \mathbb R^3 \to \mathbb R^3$ given by
$$\alpha((a,b,0),(x,y,z)) = (a,b,0)+(x,y,z)=(a+x,b+y,z)$$
The orbit of $(x,y,z)$ is given by $(a+x,b+y,z)$, where $a,b \in \mathbb R$ are arbitrary.
We can make the first two coordinates of $(a+x,b+y,z)$ whatever we want, no matter what $x$ and $y$ are, by an appropriate choice of $a$ and $b$. The only thing we can't change is the third coordinate. No matter what we chose $a$ and $b$ to be, the third coordinate of $(a+x,b+y,z)$ remains as $z$. So $z \in \mathbb R$ is an invariant of each orbit. There is a bijection between each orbit and each value of $z \in \mathbb R$.
Another way to think of it is as a quotient space. If $\alpha : G \times X \to X$ is a group action, then the quotient $X/G$ gives the orbit space. In your case $G \cong \mathbb R^2$ and $X = \mathbb R^3$, so the orbit space $G/X \cong \mathbb R^3/\mathbb R^2 \cong \mathbb R$.
