Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 w Vectors 12 2 v Vectors have magnitude and direction Jayadev S Athreya Math 241 Spring 2014 Vectors in R2 v v If two vectors have the same magnitude and direction they are equal Jayadev S Athreya Math 241 Spring 2014 Adding vectors w How do we add vectors v Jayadev S Athreya Math 241 Spring 2014 Adding vectors w Jayadev S Athreya v w v v Math 241 Spring 2014 w To form v w move to same origin form paralellogram Scaling vectors Given c R we form the scalar multiple cv by scaling magnitude by c If c 0 it flips directions 2v v v Jayadev S Athreya Math 241 Spring 2014 Vectors and Points Each vector v is equal to one with tail at the origin v v So given a point P R2 we can associate to it the vector OP where O is the origin We think of vectors and points as different Jayadev S Athreya Math 241 Spring 2014 Components v v1 v2 we call v1 v2 the components of v v2 v v1 Jayadev S Athreya Math 241 Spring 2014 Arithmetic with components v v1 v2 w w1 w2 v w v1 w1 v2 w2 v2 w 2 w2 v2 w w v v w1 Jayadev S Athreya v1v1 w1 Math 241 Spring 2014 Arithmetic with components v v1 v2 cv cv1 cv2 2v2 2v v2 v v1 Jayadev S Athreya Math 241 Spring 2014 2v1 Properties of Vectors u v w vectors in R2 c d R scalars Zero vector 0 0 0 v w w v u v w u v w v 0 v v v 0 c v w cv cw c dv cd v 1v v Jayadev S Athreya Math 241 Spring 2014 Magnitude v v1 v2 we call v1 v2 the components of v Magnitude of v is given by q kvk v12 v22 v2 v v1 What theorem are we using Jayadev S Athreya Math 241 Spring 2014 Vectors in R3 Rn Can do exactly the same things in R3 and in fact in Rn Can add scalar multiply etc Magnitude of v v1 vn is q kvk v12 v22 vn2 Jayadev S Athreya Math 241 Spring 2014 Multiplying vectors Dot product 12 3 Given v w vectors in Rn can form their dot product which is a scalar v v1 vn w w1 wn v w v1 w1 v2 w2 vn wn For example if v 2 7 4 w 5 1 3 v w 2 5 7 1 4 3 10 7 12 29 Jayadev S Athreya Math 241 Spring 2014 Properties of dot product u v w vectors in Rn c R a scalar u v w u v u w cu v c u v u v v u Jayadev S Athreya Math 241 Spring 2014 w Angle and Dot Product v v w kvkkwk cos Jayadev S Athreya Math 241 Spring 2014 Law of Cosines v w w v Law of cosines kw vk2 kwk2 kvk2 2kwkkvk cos Jayadev S Athreya Math 241 Spring 2014
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