Vectors in a Plane and  Space
     Vector product or cross product
      Vector product
         Right-handed system
         An example for the vector product in physics
         The condition for two vectors to be parallel
         The vector products of the standard unit vectors
         The vector product properties
         The vector product in the component form
      The vector product and the mixed product use, examples
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 | sinjwhere 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 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 vector or cross 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 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,
Vectors in 2D and 3D Contents
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