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Rog_Sec_12_3

# Rog_Sec_12_3 - Math 20C Lecture Examples Section 12.3 The...

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(7/24/10) Math 20C. Lecture Examples. Section 12.3. The dot product and angles between vectors Definition 1 The dot product of vectors v = ( a 1 , b 1 ) and w = ( a 2 , b 2 ) in a coordinate plane is the number v · w = a 1 a 2 + b 1 b 2 . If v = ( a 1 , b 1 , c 1 ) and w = ( a 2 , b 2 , c 2 ) are in xyz -space, then v · w = a 1 a 2 + b 1 b 2 + c 1 c 2 . Example 1 Calculate v · w for v = ( 6 , 2 ) and w = ( 4 , 3 ) . Answer: v · w = 18. Example 2 What is v · w for v = ( 6 , 2 , 3 ) and w = ( 4 , 3 , 6 ) . Answer: v · w = 0 The dot product satisfies the following rules for any vectors u , v , and w and any number λ : v · w = w · v ( λ v ) · w ) = v · ( λ w ) = λ ( v · w ) ( v + w ) · u = v · u + w · u v · v = | v | 2 The dot product is useful because of the next theorem. Theorem 1 If neither v nor w is the zero vector, then v · w = | v || w | cos θ (1) where θ is an angle between v and w . An angle between nonzero vectors can be found with the formula, cos θ = v · w | v || w | (2) which comes from (1) . Lecture notes to accompany Section 12.3 of Calculus, Early Transcendentals by Rogawski.

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Rog_Sec_12_3 - Math 20C Lecture Examples Section 12.3 The...

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