Page 2 - XII-CH10-VECTOR-LESSON NOTES

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9. i . i  j. j k.k 1 and i . j  j.k k. i 0 , where i, j andk are unit vectors along x-axis, y-axis




and z-axis respectively.
       






10. If  is the angle between two vectors a a i a j a k and b b i b j b k , then
3
2
3
2
1
1
  a b  a b 
3 3
cos  a. b  1 1 2 2 a b .
  2 2 2 2 2 2
|a||b| a  a  a 3 b  b  b 3
1
1
2
1
       
11. If a  b then a.b 0 1 2 b b c c 0 where a a i b j c k and



 a a
1 2
1
1 2
1
1
   



b a i b j c k .
2
2
2
      
12. The cross product or vector product of two vectors a and b is given by a b |a||b|sin n   ,
  
where  is the angle between a and b and n is a unit vector perpendicular to the plane of
   
a and b and +ve for a right handed rotation from a to b .
 

    |a b|  

13. |a b| |a| |b|sin and sin  , where  is the angle between a and b

  .
|a| |b|
14. Properties of cross product of vectors
   
(i) a b   

b a
            





(ii) a a  b b    i i  j j k k 0


c
c
        
(iii) i j k, j k  i andk i  
 

j
           
(iv) i j     j andk i
    i k
k
    j i, j k
       
a 0 , b 

(v) If a b    0 or a||b .
0
       



(vi) If a  a i a j a k and b  b i b j b k , then

2
3
2
3
1
1
  
i j k
 
a b  a 1 a 2 a

3
b 1 b 2 b 3
  
15. Two vectors are parallel if a b  0 .
   

16. Area if a parallelogram whose sides are represented by a and b is |a b| .
 

  a b
17. Unit vector perpendicular to a and b is .
 

|a b|
   1 

18. Area of parallelogram whose diagonals are represented by a and b is |a b| .
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