## Dot product

The dot product, also known as the scalar product or inner product, is an operation that takes two vectors and produces a scalar. Geometrically, we define the dot product of vec u and vec v with

vec u * vec v = |vec u||vec v|cos theta,

where theta is the angle between the vectors in standard position. The dot product has a few useful properties that arise from the cosine term:

If Then Therefore
vec u * vec v > 0 0º < theta < 90º acute
vec u * vec v < 0 90º < theta < 180º obtuse
vec u * vec v = 0 theta = 90º perpendicular

Algebraically, we define the dot product in RR^3 with

[u_1,u_2,u_3] * [v_1,v_2,v_3] = u_1v_1 + u_2v_2 + u_3v_3.

Essentially, we get the product of the first components, of the second components, etc., and then take the sum of the products. It works in the same way with RR^2 and any other Euclidean space.

The dot product has a few properties that are worth committing to memory. First, the dot product of a vector with itself is equivalent to the square of its magnitude:

vec a * vec a = |vec a|^2.

The dot product is commutative—order doesn’t matter:

vec a * vec b = vec b * vec a.

vec a * (vec b + vec c) = vec a * vec b + vec a * vec c.
kvec a * vec b = k(vec a * vec b) = vec a * kvec b.
vec a * vec 0 = 0.