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)

$$z\cdot w=w\cdot z .$$

2. (Additivity in the first argument)

$$(z+z')\cdot w=z\cdot w+z'\cdot w .$$

3. (Differences in the first argument)

$$(z-z')\cdot w=z\cdot w-z'\cdot w .$$

4. (Homogeneity in the first argument)

$$(\mu z)\cdot w=\mu\,(z\cdot w).$$

5. (Second argument)

$$w\cdot(z+z')=w\cdot z+w\cdot z',\qquad w\cdot(z-z')=w\cdot z-w\cdot z',\qquad w\cdot(\mu z)=\mu\,(w\cdot z).$$