I'm asked to disprove the following statement.
For vectors $\vec{x},\vec{y}\in \mathbb{R}^3:$
$(\vec{x}\cdot2\vec{y})+\vec{x}-\vec{y}=2(\vec{y}\cdot\vec{x})+\vec{x}-\vec{y}=(2\vec{x} \cdot \vec{y})+\vec{x}-\vec{y}$.
I don't see anything wrong with this though. Why is any of this wrong? I can pull the $2$ out and dot product is commutive so I am not sure why this is wrong.
Can someone explain?
Usually one considers adding vectors and scalars to be an invalid operation -- in programming terms, it doesn't "typecheck".
However, there actually are contexts, where they can be combined in "formal expressions" as "multivectors", and the whole thing does make sense. See the entire field of geometric algebra for example.
As applied to this particular statement, you can either take the first stance that "this isn't a valid expression", thus it is meaningless to ask whether it's true or false, or you can take the second stance, in which case it appears to be true, but only in a context where combining scalars and vectors this way is meaningful.