Exercise 8

Describe the algebraic properties of dot product.

• Is it right-distributive over addition: $(\vec{u}+\vec{v})\mathbin{\smash{\bullet}}\vec{w}=\vec{u}\mathbin{\smash{\bullet}}\vec{w}+\vec{v}\mathbin{\smash{\bullet}}\vec{w}$?

• Is it left-distributive (over addition)?

• Does it commute?

• Associate?

• How does it interact with scalar multiplication?

As always, you must back any assertion with a suitable argument.