Simple unit vectors orthogonal to a given vector

In geometrical computations it is often necessary to find two unit vectors such that they and a given vector form an orthogonal basis. Computationally simple forms for the two unit vectors are clearly useful. We show that they cannot always be chosen to have rational coordinates, but that in general the simplest possible vectors can be chosen to involve only one square root. We develop number-theoretic criteria for the existence of a rational vector and an effective algorithm for calculation of one if it exists. We also discuss storage and time requirements of the algorithm.