Consider three vectors a, b, and c. The scalar triple product is defined a . b ˄ c. Now, b˄c is the vector area of the parallelogram defined by b and c. So, a. b˄c is the scalar area of this parallelogram times the component of a in the direction of its normal. It follows that a . b ˄ c is the volume of the parallelepiped defined by vectors a, b, and c. This volume is independent of how the triple product is

formed from a, b, and c, except that

   (1.1)

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

   (1.2)

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

   (1.3)

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

  (1.4)

The scalar triple product is zero if any two of a, b, and c are parallel, or if a, b, and c are co-planar. If a, b, and c are non-coplanar, then any vector r can be written in terms of them:

 (1.5)

Forming the dot product of this equation with b ˄ c then we obtain

 (1.6)

so

 (1.7)

Analogous expressions can be written for β and γ. The parameters α, β, and γ are uniquely determined provided a . b˄c ≠ 0; i.e., provided that the three basis vectors are not co-planar.