21 Oct 2001

ngle Between 2 Vectors

Let a and b be two non-zero vectors represented by OA and OB respectively. The angle between a and b is defined to be the angle between OA and OB, i.e. Ð AOB.

Note that 0° £ Ð AOB £ 180°.

efinition of Scalar Product

The scalar product of two non-zero vectors a and b, denoted by a.b, is defined as

 
a.b = |a||b|cos q,  where q = angle between a and b. 

mportant Properties

  1. a.b = 0 Û a ^ b (if a ¹ 0, b ¹ 0)
  2. |a.b| = |a||b| Û a // b (if a ¹ 0, b ¹ 0)
  3. a.b = b.a
  4. a.(b + c) = a.b + a.c
  5. (la).b = l(a.b) = a.(lb) where l Î R.
  6. a.a = |a|2
Note: a.b = a.c does not imply b = c.

For the 3 mutually perpendicular unit vectors i, j, k, we have
 

i.i = j.j = k.k = 1
i.j = i.k = j.k = 0

calar Product in Cartesian Form

Let a = a1i + a2j + a3k, b = b1i + b2j + b3k.  Then, using the properties of scalar product, we have

 a.b = (a1i + a2j + a3k).(b1i + b2j + b3k) 
  = (a1i + a2j + a3k).(b1i) + 
    (a1i + a2j + a3k).(b2j) +
    (a1i + a2j + a3k).(b3k)
  = a1b1 + a2b2 + a3b3
Hence
 
    
æ a1 ö æ b1 ö    
  ç a2 ÷. ç b2 ÷  =  a1b1 + a2b2 + a3b3 
è a3 ø è b3 ø    
 

pplications of Scalar Product

1.    To prove any two non-zero vectors are perpendicular
 

a.b = 0 Û a ^ b

2.    To find the angle between two non-zero vectors.
 

   
    a.b
cos q  =  ----
    |a||b|
 

3.    To find the projection of one vector on another.
 

Length of projection of a on b = |a.b^|

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