Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Regard Euclidean space $\mathbb{R}^n$ as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the sum $z+z'$ of points and the scalar multiple $\mu z$, and write $z-z'$ and $z\cdot z'$ for the difference and the dot product.

Then for all $z,z',w\in\mathbb{R}^n$ and every $\mu\in\mathbb{R}$ the following hold.

1. (Symmetry)

2. (Additivity in the first argument)

3. (Differences in the first argument)

4. (Homogeneity in the first argument)

5. (Second argument)