1. If    = 2i - 3j + 5k and    = i - 4j + 2k then find A × B.
Sol: Given     =  2i -  3j  +  5k
                      =   i   -  4j  +  2k  
       
      ×   =  i(-6  +  20)  -  j(4 - 5)  +  k(-8  +  3)
   

  ×   =  14i  +  j  -  5k
 

2. Find the unit vector perpendicular to both vectors 2i + j + 3k and i - 2j + k.
Sol: Given vectors   = 2i + j + 3k;   = i - 2j + k

5. Find Sine of the angle between the two vectors 2i - j + k and 3i + 4j - k.

   
6. Find the area of the parallelogram with adjacent sides - i  +  k and 2j  -  k.

8. If (1, 2, 3), (2, 5, -1) and (-1, 1, 2) are the vertices of triangle then find its area.

12. Find the cartesian equation of the plane passing through the points (1, 2, 3), (2, - 3, 1) (3, 1, - 2)
Sol: Given Points     (x₁, y₁, z₁)    = (1,   2,   3)
                                   (x₂, y₂, z₂)   = (2,  - 3,  1)
                                   (x₃, y₃, z₃)   = (3,  1,  - 2)


   ( x  -  1 )( 25  -  2 )  -  ( y  -  2 )( -5  +  4 )  +  ( z  -  3 ) ( -1  +  10 )  =  0
   23x   -   23   +   y   -   2   +   9z   -   27   =   0
   23x   +   y   +   9z   -   52   =   0 

Writer Sayyad Anwar