Existence of vector fields Does there exists two vector fields $X$ and $Y$ on $\mathbb R^2$ such that the following are satisfied?

  
*
  
*$X(0)= Y(0)= 0$, where $0\in \mathbb R^2$ and for others points
  $q\in \mathbb R^2$, we have $X(q)\neq 0, Y(q)\neq 0$.
  
*For any curve $\gamma\in \mathbb R^2$, we have $\langle
 X'(t),Y(t)\rangle\geq 0$.
  
*For any curve $\gamma\in \mathbb R^2$, we have $[X'(t),Y(t)]=0$.

What happen if we change condition $3$ to $[X(t),Y(t)]=0$?
Clarification:  As $X$ is a vector field on $\mathbb R^2$, then for any path $\gamma :[0,1]\to \mathbb R^2$, we have a map $\tilde{X}(t):[0,1]\to \mathbb R^2$ by $\widetilde{X}(t)= X(\gamma(t)$,  In above question, I mean $X(t)$ by $\widetilde{X}(t)$ and $X'(t)$ is derivative of this map which can be identified with an element in $\mathbb R^2$.

I think $ X(x,y)=(x^2+y^2)\frac{\partial}{\partial y} $ and $ Y(x,y)=(x^2+y^2)\frac{\partial}{\partial x} $ should work. 
