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Unformatted text preview: Math 200, Spring 2010 Handout 2 Section 12-3 Dot Product and the Angle Between Vectors Section 12-4 The Cross Product Definition of Dot Product (or Scalar Product or Inner Product) If 1 2 3 , , a a a = a and 1 2 3 , , b b b = b , then the dot product of a and b is the number a b given by 1 1 2 2 3 3 a b a b a b = + + a b In two dimensions, 1 2 1 2 1 1 2 2 , , a a b b a b a b = = + a b 1. Find a b . (a) 2,3 , 3,7 = = a b (b) 4 3 , 2 4 6 = = + a j k b i j k Properties of the Dot Product If a , b , and c are vectors in 3 V and is a scalar, then 1. 2 a a= a Relation with Length 2. = a b b a Commutativity 3. ( ) = a b+c a b+a c Distributive Law 4. ( ) ( ) ( ) = a b a b =a b Pulling out scalars 5. = 0 a 2. Simplify the expression ( ) ( ) 2 + + v w v w v w . Theorem The angle between two vectors is chosen to satisfy 0 . If is the angle between the nonzero vectors a and b then cos = a b a b or cos = a b a b 1 cos = a b a b 3. Find a b if 3, 6 = = a b , the angle between a and b is 45 . 4. Find the angle between the vectors 4,0,2 and 2, 1,0 = = a b . Orthogonal Vectors Two nonzero vectors a and b are called orthogonal (perpendicular) if the angle between them is 2 = . Two vectors a and b are orthogonal if and only if = a b Interpretation of Dot Product We can think of a b as measuring the extent to which a and b point in the same direction. Let be the angle between nonzero vectors a and b . If 2 < , then > a b . (acute angle between vectors) If 2 = , then = a b . (orthogonal vectors) If 2 > , then < a b . ( obtuse angle between vectors) If = , then...
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