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Unformatted text preview: 46 2. VECTOR ALGEBRA 2.1 Introduction : We have already studied two operations ‘addition’ and ‘subtraction’ on vectors in class XI. In this chapter we will study the notion of another operation, namely product of two vectors. The product of two vectors results in two different ways, viz., a scalar product and a vector product. Before defining these products we shall define the angle between two vectors. 2.2 Angle between two vectors : Let two vectors a → and b → be represented by OA → and OB → respectively. Then the angle between a → and b → is the angle between their directions when these directions both converge or both diverge from their point of intersection. Fig. 2. 1 Fig. 2. 2 It is evident that if θ is the numerical measure of the angle between two vectors, then 0 ≤ θ ≤ π . 2.3 The Scalar product or Dot product Let a → and b → be two non zero vectors inclined at an angle θ . Then the scalar product of a → and b → is denoted by a → . b → and is defined as the scalar   a →   b → cos θ . Thus a → . b → =   a →   b → cos θ = ab cos θ Note : Clearly the scalar product of two vectors is a scalar quantity. Therefore the product is called scalar product. Since we are putting dot between a → and b → , it is also called dot product. O A B a b O A B a b a b θ a b θ 47 Geometrical Interpretation of Scalar Product Let OA → = a → , OB → = b → Let θ be the angle between a → and b → . From B draw BL ⊥ r to OA . OL is called the projection of b → on a → . From ∆ OLB , cos θ = OL OB Fig. 2.3 ⇒ OL = ( OB ) (cos θ ) ⇒ OL =   b → (cos θ ) … (1) Now by definition a → . b → =   a →   b → cos θ =   a → ( OL ) [ ‡ using (1)] ∴ a → . b → =   a → [ ] projection of b → on a → Projection of b → on a → = a → . b →   a → = a →   a → . b → = a ∧ . b → Projection of a → on b → = a → . b →   b → = a → . b →   b → = a → . b ∧ 2.3.1 Properties of Scalar Product : Property 1 : The scalar product of two vectors is commutative (i.e.,) a → . b → = b → . a → for any two vectors a → and b → Proof : Let a → and b → be two vectors and θ the angle between them. a → . b → =   a →   b → cos θ … (1) ∴ b → . a → =   b →   a → cos θ a b θ O L A B a b θ O L A B 48 b → . a → =   a →   b → cos θ … (2) From (1) and (2) a → . b → = b → . a → Thus dot product is commutative. Property 2 : Scalar Product of Collinear Vectors : (i) When the vectors a → and b → are collinear and are in the same direction, then θ = 0 Thus a → . b → =   a →   b → cos θ =   a →   b → (1) = ab … (1) (ii) When the vectors a → and b → are collinear and are in the opposite direction, then θ = π Thus a → . b → =   a →   b → cos θ =   a →   b → (cos π ) … (1) =   a →   b → ( − 1) = − ab Property 3 : Sign of Dot Product...
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 Spring '09
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 Vectors, Vector Space, Dot Product

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