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. AOB.

Note that 0 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 a ^ b (if a 0, b 0)
2. |a.b| = |a||b| a // b (if a 0, b 0)
3. a.b = b.a
4. a.(b + c) = a.b + a.c
5. (la).b = l(a.b) = a.(lb) where l � 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
���
 � a1 � � b1 � � � � � a2 �. � b2 � �=� a1b1 + a2b2 + a3b3� � a3 � � b3 � � �

### pplications of Scalar Product

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

 a.b = 0 � 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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