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KIC000789 - I Angle Between Two Vectors 6.23 Dot...

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Unformatted text preview: ; _ . I Angle Between Two Vectors _ 6.23 Dot ITTOdllCthvecmrs '3 If 6 is the angle between the nonzero vectors u and v, . 2; then c056: ".1, and 9=cos"[fl] 0 ; lulvl lullvl — - a! Dot Product I Ex. 3 Finding the Angle Between Vectors 1 The d“ product or inner product 0f" = (""1”) and v = (v, ,vz) ' Find the angle between the vectors u = (3,2) and v = (1,0). Ii. isu-v=uv+uv. "é um. __.,_-> 11 2 2 3 V1 %p {3 V A. 7 ctr-v" 05 0 — 5 Properties of the Dot Product \, C M - I w Let u, v, and w be vectors and let c be a scalar. ‘3 I 7 : 1. u-v=v-u ._ M a. s 2- u-u=lul2 _ A: a» '3: W“ 3 3'. 00:0 , r: .. “333 u-V:(3.Qt. Z we " é . 4. u.(v+wFu,v+u.w_ “1+"? .w—_'_u.w+v.w 2 E 3' ‘1 J 3.5-}! ' i ' (,6; G- 929' 1 C09- [ (f) 6 :- 93,696 5. (cu).-Vf~=u-.(cv)=c(u-v) 3.; Orthogonal Vectors ’ i r .s . . =; The vectors u and v are orthogonal 1f and only If u v — 0 i EX. 1 Findingthe Dot Produfl _ 3 3 Flnd the i . i: i ‘3 a 4 (kit product. . , _ , , 3.1; Ex. 4 Proving Vectors are orthogonal a. 3-~—_—"'Ti" _ , < (i I) ( 1’ 2) ' - ' ' " i Prove that the vectors u = (2,3) and V — ('6’4) are orthogonal 1/ ) +X’Z)‘ H :LLQQ +304} 53 3-H: 3421—0. 2. . .', ,”.'-4L0> 3% . b. (21—})-<31—S}) : 31:49" :0 3.: 30"403”) -(43-,o*>—~£.o‘€>) Ul—V w Ovtflwl ”‘13-‘>-'<??t>' =24 MW l: i E 3 ' -_‘::: ‘10” m E . 33 3: 3 . ' 3 Uzezthufing 139* PfotlucttoFlld Length :‘ Pm. t' f and v 3. 3 3.‘ E 3 j _ 3 3 3“, we OtiPISdlm‘o-fmd the bum Offl‘le vector u = (4, -3). 3333 lee 1011 0 u 3 3 tov is Proj “1 E v- 3 E U U 3 l t _ ' ‘ :33 i If" and v are nonzero vectors, the projecuon Of,” on v M: I? 0'54“”) 1 r L 7 33 G~Ve¢H,-z‘>‘}é‘4’j?>« . z; -‘IWC‘Q? :33 1%.": 7. _ if? v 33: E; HAL. 2s 331 ML: luc- S 3 l . TEE- ©lDrJg Tanple F , ' , 3 % 1 © Dr. Jo Temple, Fall 2008 197 33 3 E Precafwluhmrgégg? 33 3 3-3=-3% 3!} Precalculus—AMTH I550 .Siiini ...
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