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


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

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

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

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

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

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

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

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