  In three-dimensional Euclidean geometry, the scalar product produces a number
associated to two vectors, while the vector product computes a vector
perpendicular to them. These are key tools of physics, chemistry and
engineering and supported by a rich vector calculus of 18th and 19th century
results. This paper extends this calculus to arbitrary metrical geometries on
three-dimensional space, generalising key results of Lagrange, Jacobi, Binet
and Cauchy in a purely algebraic setting which applies also to general fields,
including finite fields. We will then apply these vector theorems to set up the
basic framework of rational trigonometry in the three-dimensional affine space
and the related two-dimensional projective plane, and show an example of its
applications to relativistic geometry.
