Properties of Dot Product

Theorem
Let $\vec u, \vec v, \vec w$ be vectors in the vector space $\R^n$.

Let $c$ be a real scalar.

The dot product has the following properties:


 * $\vec u \cdot \vec u \ge 0; \vec u \cdot \vec u = 0 \iff \vec u = \vec 0$.


 * $\vec u \cdot \vec v = \vec v \cdot \vec u$.


 * $(\vec u + \vec v) \cdot \vec w = \vec u \cdot \vec w + \vec v \cdot \vec w$.


 * $c \vec u \cdot \vec v = c (\vec u \cdot \vec v)$.

Proof
From the definition of dot product


 * $\displaystyle \vec a \cdot \vec b = \sum_{i-1}^n a_i b_i$

Proof that $\vec u \cdot \vec u \ge 0$:

$\displaystyle \vec u \cdot \vec u = \sum_{i-1}^n u_i^2 \ge 0$.

Proof that $\vec u \cdot \vec u = 0 \iff \vec u = \vec 0$:

Let $\vec u \cdot \vec u = 0$.

Then $\displaystyle \sum_{i-1}^n u_i^2 = 0$ and so $\forall i: u_i = 0$.

Now suppose $\vec u = \vec 0$.

Then $\displaystyle \sum_{i-1}^n u_i^2 = 0$ and so $\vec u \cdot \vec u = 0$.