2.13. The Scalar Triple Product

The scalar triple product is given by

        A( B×C )= ε ijk a i b j c k   =| a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 |

                        = volume of parallelepiped with sides A, B and C.

Theorem 2.14

3 vectors A, B and C in R3 are linearly dependent  Û  A( B×C )=0 .

Proof

Þ

If B and C are linearly dependent, then B×C=0  and theorem is proved.

If B and C are L.I., then there exists α,β,γ , not all zero, such that

        αA+βB+γC=0

Furthermore α0  since otherwise α=β=γ=0 .  Taking ( B×C )  on both sides gives  αA( B×C )=0 .  QED.

Ü

Given A( B×C )=0 , we have either B×C=0  or A is orthogonal to B×C0 .  For the former case, B and C are dependent and the theorem is proved.  For the latter case, theorem 2.13 says that B, C and B×C  form a basis so that

        A=αB+βC+γB×C

Taking ( B×C )  on both sides gives 0=γ B×C 2  so that γ=0 , i.e.,

        A=αB+βC

QED.