Chapter 3 Vectors Q: 3, 7 P: 1, 4, 5, 13, 19, 30, 35, 47, 63

Transcription

1 Chpter 3 Vectors Q: 3, 7 P: 1, 4, 5, 13, 19, 30, 35, 47, 63

2 Vector A vector s quntt tht hs mgntude nd drecton. Notton, dfferent ws we cn wrte vectors A = A A unt vector notton A = (A, A ) vector s components A = (A, ) mgntude nd drecton (4, 3) A = 4 3 A = (4, 3) A = (5, 37 0 ) A = A cos A = A sn A = 2 2 A A

3 Vector propertes: - Eqult of vectors, the 2 vectors hve the sme mgntude nd drecton - Addng vectors, cn onl dd the sme unts - Add them up geometrcll(pctures) - Add them up lgercll(mth) - Vectors oe rules of lger: - communctve lw - ssoctve lw - to sutrct vectors ( ) c ( c) d ( )

4 Components of vectors = cos = sn sn (4, 3) cos = tn 2 2

5 Addng vectors Vectors re cn dded up usng grphcl (geometrc) method. The cn lso e dded ddng ther le components (lgerc) method Emple of grphcl method: B B A A

6 12 C 7 B 7 B 5 A 5 A

7 Our new vector ws C = (5, 12) 2 2 It hs mgntude of C = = 13 Let s fnd the ngle tht C mes wth the -s. tn = 12/5 tn -1 (12/5) = 12 C = 67 degrees Could lso do: sn -1 (12/13) = cos -1 (5/13) =

8 When ddng vectors usng the components, the mportnt rule s tht ou cn onl dd smlr components to ech other. Onl -components cn e dded to -components. Onl -components cn e dded to -components. r r r r r r ) ( ) ( ) ( ) (

9 Negtve of Vector When ou hve mnus sgn n front of vector, ll ou hve to do s flp the vector to the opposte drecton. A -A B -B C -C

10 Sutrctng Vectors When dong A B, thn of t s ddng A to Negtve B. A B = A + (-B) A A B -B

11 Cn Add More thn 2 vectors. A B C D

12 Vectors nd the Lws of Phscs You ve notced now tht we hve lws een pcng coordnte sstem wth to the rght nd vertcll upwrd. Ths loos nce nd net ut tht s the onl reson. We cn choose coordnte sstem tht s rotted through n ngle nd fter solvng phscs prolem get the sme results. The lws of phscs re ndependent of the coordnte sstem. So pc the coordnte sstem tht s the most convenent.

13 Multplng Vectors Multpl vector sclr. Dong so gves us new vector. m s vector m s found multplng the sclr ech component of the vector let s = 5 nd 4 6 m If s s postve, the two vectors re n the sme drecton. If s s negtve, the two vectors re n opposte drectons. Multplng -1 chnges vector s drecton

14 There re two ws to multpl vector nother vector. The sclr product. Also clled the dot product. cos nd re the mgntudes of the vectors nd s the ngle etween them. The dot product gves sclr s result. It s relted to the proecton of one vector onto the other. Put fg on the ord.

15 Another w to fnd the dot product. So f ou now the components of the two vectors, ou cn fnd the dot product wthout nowng the ngle n etween vectors. Ths s useful n fndng the ngle. The dot product s communctve: If ou hve the vectors n unt vector notton use: These re true ecuse the unt vectors re orthogonl to ech other. 0 1

16 The other w to multpl two vectors s the vector (cross) product. Ths produces thrd vector tht s perpendculr to the other two vectors. c c sn Where, nd re the mgntudes of the two vectors eng multpled nd q s the smller ngle etween the vectors. The cross product s relted to the se of the prllelogrm tht s formed the two vectors. The re s equl n mgntude to the mgntude of the cross product. The cross product vector s perpendculr to the prllelogrm. The cross product s NOT communctve. sn

17 If ou hve the vectors n unt vector notton, use the rules: c 0

18 Rght hnded sstem When the s re set up so tht ou hve rght hnded coordnte sstem. Ths goes long wth usng the rght hnd rule to fnd the drecton of the cross product. nde fnger frst vector mddle fnger second vector thum cross product or Pont fngers n drecton of frst vector. Curl them towrds the second vector. Thum wll e n drecton of cross product.

19 Usng mtr to fnd cross product The vertcl rs round the mtr represents the determnnt of the mtr. ) ( ) ( ) (

20 Just le erler tht we could use the two defntons of the dot product to fnd the ngle n etween vectors, we cn do smlr process wth the cross product to fnd the ngle n etween vectors. As pproches ero, the dot product s ecomes. As cos pproches ero, the cross product s ecomes ero. sn As pproches 90 0, the dot product s ecomes ero. As pproches 90 0, the cross product s ecomes.

21 Prolem 39 Fnd the ngle etween the two vectors: Prolem 40 A B C Fnd 3C (2A B)