12.3 The Dot Product

# 12.3 The Dot Product - Math 224 Calculus III 12.3 The Dot...

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Math 224 – Calculus III 1 12.3 The Dot Product Recommended Homework: # 1-47, 51, 55(odds) In 12.1 we saw addition, subtraction and scalar multiplication of vectors. Here we look at “multiplying” two vectors. We actually define two types of product: (i) The Dot Product (ii) The Cross Product (section 12.4) The Dot Product (or Inner Product) Given vectors 1 2 3 ,, a a a a and 1 2 3 b b b b in component form their dot product is given by: 1 1 2 2 3 3 a b a b a b ab Note : The result of a dot product is a real number ! The dot product of two vectors does not have a convenient interpretation, but it is related to the angle between the two vectors. The equivalent rule also works for vectors in 2 Properties of the Dot Product The familiar arithmetic properties (commutative, associative etc) hold for the dot product (try them!) Given the vectors a , b and c and suppose c is a scalar. (i) 2 a a = a (ii) a b = b a (iii) ( a b +c) = a b a c (iv) (( c c c ( a) b = a b) = a b) (v) 0 0 a = Geometric Interpretation of the Dot Product : Theorem : If is the angle between vectors a , b

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12.3 The Dot Product - Math 224 Calculus III 12.3 The Dot...

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