From 2-D to Higher Dot Products

Extending dot products to vectors in 3 or more dimensions is extremely simple.

The geometric definition stays exactly the same, since it does not depend on the number of dimensions of the coordinate space. Let $\theta$ be the angle between $\vec{a}$ and $\vec{b}$ when the two vectors are positioned with their initial points coinciding. We know that the following holds: $$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|cos\theta$$

The algebraic definition for arbitrary $n$-dimensional space is a simple generalization. Let $\vec{a}=\left< a_1,a_2,...,a_n \right>$ and $\vec{b}=\left< b_1,b_2,...,b_n \right>$. Then, the dot product is defined as follows: $$\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+...+a_nb_n$$

Using the same procedure as in 2 dimensions, one can prove that these two definitions of the dot product are equivalent. All other properties and uses of the dot product are the same; the only change is the increase in the amount of computation needed to compute and constructively use the dot product.