Vector Product  and Scalar Triple Product                    

 

 

Example:   Evaluate the vector product,  U x V  for the vectors  

 

              U   =   3 i    +   4 j   +   5 k     and     V   =    i    -   2 j   -   3 k    

 

 

Use the “right hand rule” to rotate the vector U into V.  The resultant of this product

is a new vector, W,  perpendicular to U and V. 

 

 

Strategy:   Use a  3 x 3  determinant (det) to  calculate the vector product of U and V 

with  i, j, and k in the first row,  U   in the second row and   V   in the third row.

 

 

                                                       i            j          k

 

     W  =       U   x   V   =   det         3            4           5  

                            

                                                       1           -2         -3

 

 

                                4    5                      3     5                          3     4

  U   x   V   =  i det                   -  j det                        + k det     

                              -2   -3                      1    -3                         1    -2

 

 

     U   x   V   =     i  [ (4)(-3) – (-2)(5)]  -  j  [ (3)(-3) – (1)(5)]  +  k  [ (3)(-2) – (1)(4)] 

 

    W   =     - 2 i    +  14 j    -  10 k            (result for vector product)

 

 

Note:  W  is a new vector that is perpendicular to vectors  U   and   V   so the

dot product of   W  with both  U  and   V  should be zero.

 

 

Check:         W ·  U  =  ( - 2 i    +  14 j    -  10 k )  ·  ( 3 i    +   4 j   +   5 k  )       

 

        W ·  U  =  ( - 2)(3)  + (14)(4) + (- 10)(5)  =  0

 

Also  W ·  V  =  ( - 2)(1)  + (14)(-2) + (- 10)( -3)  =  0

 

 

 

Click here to continue with vector products.

 



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