Scalar Triple Product

(A.53) |

So, the ``volume'' is positive if , , and form a right-handed set (i.e., if lies above the plane of and , in the sense determined from a right-hand circulation rule by rotating onto ), and negative if they form a left-handed set. The triple product is unchanged if the dot and cross product operators are interchanged,

(A.54) |

The triple product is also invariant under any cyclic permutation of , , and ,

(A.55) |

but any anti-cyclic permutation causes it to change sign,

(A.56) |

The scalar triple product is zero if any two of , , and are parallel, or if , , and are coplanar.

If , , and are non-coplanar then any vector can be written in terms of them: that is,

(A.57) |

Forming the dot product of this equation with , we then obtain

(A.58) |

so

(A.59) |

Analogous expressions can be written for and . The parameters , , and are uniquely determined provided : that is, provided the three vectors are non-coplanar.