Model for the Vector Product, Distributive Law by Richard P. Baker, Baker #155

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over 100 of his models are in the museum collections.

The typed part of a paper label on the edge of the wooded base reads: “No. 155 (/) Vector product; distr.law.” The number 55 is written twice on the base as are three equations, [(A+B)]C=OD”, [AC]=OA”, and [BC]=OB”. In these equations Baker is representing the vectors on the left hand side by a single upper case letter that represents the vector emanating from the point labeled O and ending at the point labeled by that letter. On the right-hand side of these equations Baker represents that same type of vector but shows the O explicitly. It should be noted that since the single upper case letters are vectors, the product means the cross product.

Vectors A, B, and A+B, all lie on the base. Vector C is the vertical metal arrow and, therefore, is perpendicular to A, B, and A+B, which is the diagonal of parallelogram OADB, i.e. the vector D. Because the cross product of two vectors is perpendicular to both vectors, the cross products (A+B)C=DC, AC, and BC, are all perpendicular to C so also lie on the wooden base.

The model illustrates a special case of the distributive law for the cross product of vectors, i.e., when the vectors that form the sum (here A and B) define a plane that is perpendicular to the vector being distributed over the sum (here C). In this case the parallelograms OADB and OA”D”B” are similar where A”, B” and D” are defined in the original equations. Finally, if we let U denote the unit vector in the same direction as C then [(A+B)]U=D’, [AU]=A’, and [BU]=B’ and OADB and OA’D’B’ are congruent.

While there are also vertical wire pieces marking the points A, B, and D, it is not clear how the height of these wire pieces relate to the vectors.

Currently not on view