Vector Analysis · 1.1
Physics
Problem
Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive, a) when the three vectors are coplanar; b) in the general case.Intuitive explanation
The key is to use the geometric definitions of dot and cross products—not component formulas—and leverage area and projection arguments. For coplanar vectors, you can draw everything in 2D and show that distributing the operation (whether dot or cross) before or after combining vectors gives the same result by comparing projections or areas. In the general case, extend this by decomposing vectors into components parallel and perpendicular to a reference direction, then apply the coplanar result to each piece—the distributive property follows because it holds for every geometric "slice" of the configuration.
Intuitive explanation
The key is to use the geometric definitions of dot and cross products—not component formulas—and leverage area and projection arguments. For coplanar vectors, you can draw everything in 2D and show that distributing the operation (whether dot or cross) before or after combining vectors gives the same result by comparing projections or areas. In the general case, extend this by decomposing vectors into components parallel and perpendicular to a reference direction, then apply the coplanar result to each piece—the distributive property follows because it holds for every geometric "slice" of the configuration.