Vectors in a Plane and  Space The mixed product or the scalar triple product
The mixed product or scalar triple product definition
The mixed product properties
The condition for three vectors to be coplanar
The mixed product or scalar triple product expressed in terms of components The vector product and the mixed product use, examples
The mixed product
The mixed product or scalar triple product definition
The mixed product (or the scalar triple product) is the scalar product of the first vector with the vector product
of the other two vectors denoted as  a · ( b ´ c ).
 Geometrically, the mixed product is the volume of a parallelepiped defined by vectors, a , b and c as shows the right figure. The vector b ´ c is perpendicular to the base of the  parallelepiped and its magnitude equals the area of the base,  B = | b ´ c |. The altitude of the parallelepiped is projection of the  vector a in the direction of the vector b ´ c, so h = | a | · cos j. Therefore, the scalar product of the vector  a  and vector  b ´ c is equal to the volume of the parallelepiped. The sign of the scalar triple product can be either positive or negative, as   a · ( b ´ c ) = - a · ( c ´ b ).
The mixed product properties The condition for three vectors to be coplanar
The mixed product is zero if any two of vectors, a , b and c are parallel, or if a , b and c are coplanar. That is, when the given three vectors are coplanar the altitude of the parallelepiped is zero and thus the scalar triple product is zero, The mixed product or scalar triple product expressed in terms of components
The scalar triple product expressed in terms of the components of vectors, a = axi + ay j + azk,
b = bxi + by j + bzk  and  c = cxi + cy j + czk, The above formula can be derived from the determinant expanded by minors through the elements of the first row, Therefore, vectors, a, b and c will be coplanar if the determinant is zero.
The vector product and the mixed product use examples
Example:   Given are vectors, a = i - 2 k and  b = - i + 3 j + k, determine the vector  c a ´ b and the area of a parallelogram formed by vectors, a and b.
 Solution:  The area of the parallelogram, or  Example:  Vertices of a triangle are, A(-1, 0, 1), B(3, -2, 0) and C(4, 1, -2), find the length of the altitude hb.
Solution:  Using the area of a triangle, as the area of the triangle where   then, the area of the triangle, and the length of the side b, thus, the length of the altitude, Example:  Find the angles that the unit vector, which is orthogonal to the plane formed by vectors,
a = -i + 2 j + k  and  b = 3i - 2 j + 4k, makes with the coordinate axes.
 Solution:  The unit vector which is orthogonal to the plane, formed by the vectors, a and b, is the direction vector of a vector c, such that  Angles, which the unit vector c°  forms with coordinate axes, we find by using the formula, Example:  Given are vectors, a = -2i - 3 jb = -i - 2 j + 3k and c = -i + 2 j + k, find the projection (the scalar component) of the vector a onto vector d = b ´ c.
Solution:  Let first find the vector d,  Example:  Given are vectors, a = -3i + 4 j - l kb = 2i -  j + k  and  c = i - 4 j -3l k, determine the
parameter
l such that vectors to be coplanar.
Solution:  Vectors lie on the same plane if their scalar triple product is zero, i.e., V = 0, therefore vectors’
coordinates must satisfy the condition, Example:  Examine if vectors, a = 4i + 2 + kb = 3i + 3 j  - 2k  and  c = - 5i -  j - 4k, are coplanar and if so, prove their linear dependence.
Solution:  Check for coplanarity, Since vectors are coplanar each of them can be represented as linear combination of other two. Example:  Prove that points, A(-2, 1, 0), B(3, -2, -1), C(1, -1, 2) and D(-3, 2, -5), all lie on the same plane.
Solution:  If all given points belong to the same plane then vectors, AB, AC and AD are coplanar, therefore the scalar triple product (AB ´ AC) · AD = 0.  thus, all given points lie on the same plane.   Vectors in 2D and 3D Contents 