One can easily prove that
,
and so the scalar triple product may be symbolically denoted by abc.
By the properties of determinants
abc = cab = bca,
abc = - bac = - acb.
From the geometrical interpretation of the triple product it is follows that
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The triple product of coplanar vectors equals
zero, and vice versa, if the triple product of non-zero vectors
equals zero then the vectors are coplanar.
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In view of the fact that any three linear dependent vectors are coplanar, we obtain the following Corollary:
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The triple product of non-zero vectors equals
zero if and only if the vectors are linear dependent.
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