Definition:Dot Product

Definition
Given any two vectors in $$\mathbb R^n$$, $$a$$ and $$b$$, the Dot Product, or  Standard Inner Product is defined as:

$$a \bullet b = a_1b_1 + a_2b_2 + ... + a_nb_n = \sum_{i-1}^{n} a_ib_i$$

Properties Shown in $$\mathbb R^2$$ and $$\mathbb R^3$$
Let $$u, v, w$$ be vectors in the $$R^2$$ or $$R^3$$ vector space, and let $$c$$ be a real scalar. The dot product on $$R^2$$ and $$R^3$$ holds to the following properties:


 * $$u \bullet u \ge 0; u \bullet u = 0  $$ iff $$u = 0$$


 * $$u \bullet v = v \bullet u$$


 * $$(u+v) \bullet w = u \bullet w + v \bullet w$$


 * $$cu \bullet v = c(u \bullet v)$$