Vector: Outer Product and Volume

 

Motivation:

                      

                        Find the volume spanned by the three vectors as below:

 

                                   

 

                        And from previous study, we can find that the area of parallelogram produced from  and  is

 

                                       2  OAB =  

 

                      For the height of the quadrangular prism,  it  is

 

                               h = the projection length of   on normal vector of   and

 

                    Let   be the unit vector on the direction of normal vector of   and .

                    Then

                                     h = 

 

                     With height and the area of the bottom parallelogram, 

 

                                 Volume of prism = h

                                                             =  (  )

 

 

 

                    Now, we put the whole thing into a 3-dimensional coordinate system.

                     Let

                                    = (a₁, a₂, a₃ )

                                    = (b₁, b₂, b₃ )

                                    = (c₁, c₂, c₃ )

 

 

 

                                     

                                   =

 

 

                     From this, we hope that we can define a vector from  and  such that

 

                          1.  the vector is perpendicular to  and ;

                          2.  the magnitude of the vector is equal to the area of

                                 parallelogram spanned by  and

 

                     That is how we define the outer product of   and .

 

Definition ( Outer Product )

               Let  ,  ,   be three unit vectors along the positive directions of x-axis, y-axis, and z-axis of the 3 dimensional coordinate system.  If we have two vectors

 

                       = a₁  + a₂  + a₃

                        = b₁  + b₂  + b₃

 

               then we define the outer product of   and  as

 

                     =   +  +

 

               In the form of the determinant, it can be represented as

 

                     

                        =

 

              If you are not familiar with determinant, just treat is as an equivalent representation of the one above.

 

              Please notice that outer product is not commutative; in other words,

                                    

 

 

 

 

 

 

 

Theorem ( Volume of Quadrangular Prism )

 

                        The volume of the quadrangular prism shown below is

 

                                            V  = |  (  ) |

                          

 

                   In the form of determinant, it is

 

                                 V = |  |