The scalar triple product is given by
= volume of parallelepiped with sides A, B and C.
3 vectors A, B and C in R3 are linearly dependent Û .
If B and C are linearly dependent, then and theorem is proved.
If B and C are L.I., then there exists , not all zero, such that
Given , we have either or A is orthogonal to . 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 form a basis so that
Taking on both sides gives so that , i.e.,