(A.1) |

This defines

(A.2) |

However, has the same length and direction as , and, thus, represents the same vector, . Likewise, and both represent the vector . Thus, the previous equation is equivalent to

(A.3) |

We conclude that the addition of vectors is

(A.4) |

The null vector, , is represented by a displacement of zero length and arbitrary direction. Because the result of combining such a displacement with a finite length displacement is the same as the latter displacement by itself, it follows that

(A.5) |

where is a general vector. The negative of is defined as that vector which has the same magnitude, but acts in the opposite direction, and is denoted . The sum of and is thus the null vector: that is,

(A.6) |

We can also define the difference of two vectors, and , as

(A.7) |

This definition of

If is a scalar then the expression denotes a vector whose direction is the same as , and whose magnitude is times that of . (This definition becomes obvious when is an integer.) If is negative then, because , it follows that is a vector whose magnitude is times that of , and whose direction is opposite to . These definitions imply that if and are two scalars then

(A.8) | ||

(A.9) | ||

(A.10) |