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ALGEBRA
 :: Vector product or cross product

  The vector product of two vectors, a and b is the vector a ´ b perpendicular to given vectors, and the magnitude of which
| a ´ b | = | a | · | b | · sinj,  where j is the angle between a and b.

That is, the vector product a ´ b is the vector magnitude of which equals the area of a parallelogram with sides a and b, and the direction of which is perpendicular to the plane of the given vectors, forming a 
 right-handed system as is shown in the below diagram,

A right-handed system is so called because vectors, a , b and  a ´ b  have the orientation of the thumb, the forefinger, and the middle finger of the right hand.

 That is, if the thumb is placed in the direction of the first operand (a) and the forefinger in the direction of the second operand (b), then the resultant vector   (a ´ b) is coming out of the middle finger.

An example for the vector product in physics

An example for the vector product in physics is a torque (a moment of a force - a rotational force).

The force applied to a lever, multiplied by its distance from the lever's fulcrum O, is the torque T, as is shown in the diagram.

 r  = OP is the vector from the axis of rotation to the point P on which the force is acting.

The distance from the direction line of the force to O is,

 If a force of magnitude | F | acts at an angle j to the displacement arm, and within the plane perpendicular to the rotation axis, then from the definition of the cross product, the magnitude of the torque is,

The direction of the torque is determined by the vector 
product,
The vector product is not commutative, that is

The thumb of the right hand is now placed in the direction of b, the forefinger in the direction of a, and thus the middle finger points in the direction of b ´ a.

Two non-zero vectors are parallel if

that is, the angle between vectors (a, b) = 0° or 180°.

The vector product of a vector with itself or with a parallel 
vector is zero or the null vector, i.e., 

The vector products of the standard unit vectors

The vector product properties

The vector product in the component form

The above component notation of the vector product can also be written formally as a symbolic determinant expanded by minors through the elements of the first row. A minor is the reduced determinant formed by omitting the i-th row and  j-th column of a matrix, multiplied by (-1)i + j.
 :: 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 this 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:  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

Example:  Given are vectors, a = -3i + 4 j - l k,  b = 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,
 
 
 
 
 
 
 
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