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University of Toronto St. George

Mathematics

MAT237Y1

Stanzeck

Winter

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MAT237Y1a.doc The Geometry of Euclidean Space N V ECTORS IN R n u =(u ,,u ) A vector in R can be written as 1 n . Definitions u 1= v1 u = v u = u1,,u n) v =(v1,,v n) un = vn If and , then . u = u1, un), R . Some Properties n u R n n u + v R 1) v R . R n u R nu R 2) . 3) u+ v = v+u . 4) u+ (v+w )= v+u +w . u = (u , ,u ) u+ (u)= 0 =(0, ,0)=(u )+u 5) 1 n and . 6) u 0 u = 0 u . 7) u+ v )=u+v . ( + u =u+ u 8) . ( u = (u) 9) . 10)1u =u . N ORM Definition 2 2 Let u = u1 u n). The norm isu = u 1 ++u n . Properties u 0 1) . u+ v u + v 2) . u = u 3) . Page 1 of 23 www.notesolution.com MAT237Y1a.doc D OT P RODUCT Let u = u1,,u n )and v = v1,,v n), thenuv = u1 1++u v n n. Properties 1) uv = vu. 2) u+ v )w = uw + vw . (uv)= u(v ) 3) . 4) uu 0 , moreover,uu = 0 u = 0 More Properties 2 1) uu = u . 2) uv u v (Cauchy-Schwartz). 2 2 2 2 3) u+ v u+(v = u ) +2uv+ v and (u v u( v = u) 2uv+ v . u+ v u(v = u )2 v 2 4) . The Cosine Law 2 2 2 v = v + u 2 v u cos 2 2 (vu )vu = v) + u 2 v u cos v 2 2 2 2 v - u v 2u + u = v + u 2 v u cos = v u cos u Theorem uv = v u cos,u,v 0 . Some Basic Consequences 1) uv u v . 0 < 2) uv > 0means 2 . = uv =0 means 2 . uv < 0means . P ROJECTION Page 2 of 23 www.notesolution.com

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