University Mathematics Handbook (2015)

IX. Vector Analysis

Chapter 1. Vector Function, Hodograph

a.  Vector , when , , and are real functions with variable , is called vector function.

b.  For every constant , vector function defines constant vector .

c.  All operations in vector functions, such as addition, subtraction, multiplication by scalar, inner product and cross product are defined in a similar way (III.4).

d.  The locus of the ends of vectors when , which start in the origin, is the graph or path of vector function and is called hodograph.

e.  A graph of vector function can also be represented in a parametric form: , , .

f.  Example: The graph of vector function

is a curve the parametric equation of which is

Since for every constant , the end of the vector moves along a spiral situated on the side of a circular cylinder (see illustration).