I define a vector operation as follows: $$\vec{a}\tilde\times \vec{b}=\vec{a}.\vec{b}(\hat{\perp})=ab \cos\theta(\hat{\perp})$$ where $(\hat{\perp})$ is a unit vector perpendicular to $\vec{a}$ and $\vec{b}$ which we get by right hand rule.
Now can we prove whether this vector operation is distributive or not?