DSE 2012 Question 50: Vector spaces, rank and eigenvalues

Let $x=(x_{1},x_{2},\ldots,x_{n})$ and $y=(y_{1},y_{2},\ldots,y_{n})$ be two vectors in $\Re^{n}$. Define, $x\otimes y=\sum_{i=1}^{n}x_{i}|y_{i}|$, where $|y_{i}|$ stands for absolute value. How many of the following statements are correct: Statement 1: $x\otimes y=y\otimes x$. Statement 2: $x\otimes x=0$ implies $x=(0,0,\ldots,0)$. Statement 3: $x\otimes(c.y)=c(x\otimes y)$, where $c\in\Re_{++}$ and $c.y=(cy_{1},cy_{2},\ldots,cy_{n})$