The Dot Product

A certain multiplication process recurs frequently in dealing with two vectors. That process takes the product of the two vectors' x coordinates (in two dimensions, x₁·x₂), the product of their y coordinates (y₁·y₂), and then sums those individual products (x₁.x₂+y₁·y₂). In three or more dimensions, we also add on z₁·z₂, and so on. Thus, there are as many items to add as there are dimensions or axes. In mathematical shorthand, say with vectors E₁ and E₂ you'll see the process written as E₁·E₂ (pronounced "E one dot E two"). Capsulizing the general idea in symbols for two dimensions:

E₁·E₂ = x₁·x₂ + y₁·y₂.....(1)

The resulting number isn't a vector. Instead, it's simply a scalar and goes by any of three names—the scalar product, dot product, or inner product. The dot, in other words, symbolizes multiplication whether we're multiplying scalars (pure numbers) or vectors.
As an example, suppose we've got two vectors in three-dimensional space. The first vector is 4,1,3 (i.e. the vector going from the graph's origin to the point at x=4, y=1, and z=3). The second vector is 2,-3,2. The dot product then is (4,1,3)·(2,-3,2). It's computed (Eq. 5.4 for three dimensions) as x1·x2 +y1·y2 +z1·z2 = (4·2)+(1·[-3])+(3·2)=8-3+6=11.