The dot product is a vector operation, which, along with the cross product and scalar multiplication, comprise the three types of "multiplication" operations that can be performed on vectors. Vector division is not defined.

The fundamental definition of a dot product is the product of the scalar magnitudes (lengths) of each vector and the cosine of the angle in between them. \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\theta)

This definition can be equivalently written as a scalar projection, or component, of vector u onto vector v, times the magnitude of vector v: \vec{u} \cdot \vec{v} = |\vec{v}| \mathrm{comp}_{\vec{v}} \vec{u}

By consequence of the definition, a dot product is the sum of the products of corresponding elements within the vector. This is often regarded as an alternative definition. Given the two (3-dimensional) vectors u and v:

  • \vec{u} = \langle u_1,u_2,u_3 \rangle = u_1 \mathbf{\hat{i}} + u_2 \mathbf{\hat{j}} + u_3 \mathbf{\hat{k}}
  • \vec{v} = \langle v_1,v_2,v_3 \rangle = v_1 \mathbf{\hat{i}} + v_2 \mathbf{\hat{j}} + v_3 \mathbf{\hat{k}}

The dot product is:

\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3