operators to three vectors. The scalar triple product is sometimes denoted by (a b c) and defined as: It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are right-handed. In components (with respect to a right-handed orthonormal basis), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows The scalar triple product is linear in all three entries and anti-symmetric in the following sense: Conversion between multiple Cartesian bases All examples thus far have dealt with vectors expressed in terms of the same basis, namely, the e basis {e1, e2, e3}. However, a vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same vector. In the e basis, a vector a is expressed, by definition, as The scalar components in the e basis are, by definition, In another orthonormal basis n = {n1, n2, n3} that is not necessarily aligned with e, the vector a is expressed as and the scalar components in the n basis are, by definition, The values of p, q, r, and u, v, w relate to the unit vectors in such a way that the resulting vector sum is exactly the same physical vector a in both cases. It is common to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle). In such a case it is necessary to develop a method to convert between bases so the basic vector operations such as addition and subtraction can be performed. One way to express u, v, w in terms of p, q, r is to use column matrices along with a direction cosine matrix containing the information that relates the