It is convenient to define unit vectors, , , and , parallel to , , and , respectively. Incidentally, a unit vector is a vector whose magnitude is unity. The position vector, , of some general point whose Cartesian coordinates are ( , , ) is then given by

(A.11) |

In other words, we can get from to by moving a distance parallel to , then a distance parallel to , and then a distance parallel to . Similarly, if is an arbitrary vector then

(A.12) |

where , , and are termed the

According to the three-dimensional generalization of the Pythagorean theorem, the distance is given by

(A.13) |

By analogy, the magnitude of a general vector takes the form

(A.14) |

If and then it is easily demonstrated that

(A.15) |

Furthermore, if is a scalar then it is apparent that

(A.16) |