home *** CD-ROM | disk | FTP | other *** search
| Unknown | 1997-04-17 | 7.7 KB |
open in: 
MacOS 8.1
|
Win98
|
DOS
view JSON data
|
view as text
This file was not able to be converted.
This format is not currently supported by dexvert.
| Confidence | Program | Detection | Match Type | Support
|
|---|
| 1%
| dexvert
| Eclipse Tutorial (other/eclipseTutorial)
| ext
| Unsupported |
| 1%
| dexvert
| JuggleKrazy Tutorial (other/juggleKrazyTutorial)
| ext
| Unsupported |
| 100%
| file
| data
| default
| |
| 100%
| gt2
| Kopftext: 'TUTOR 06'
| default (weak)
|
|
hex view+--------+-------------------------+-------------------------+--------+--------+
|00000000| 54 55 54 4f 52 20 30 36 | f3 1d 00 00 d6 00 00 00 |TUTOR 06|........|
|00000010| 53 65 63 74 69 6f 6e 20 | 38 2e 33 20 20 56 65 63 |Section |8.3 Vec|
|00000020| 74 6f 72 73 20 69 6e 20 | 74 68 65 20 50 6c 61 6e |tors in |the Plan|
|00000030| 65 0d 0a 00 46 6f 72 20 | 6d 6f 72 65 20 70 72 61 |e...For |more pra|
|00000040| 63 74 69 63 65 3a 0d 0a | 00 0d 0b 00 20 20 20 20 |ctice:..|.... |
|00000050| 20 10 61 38 2d 33 2d 33 | 0e 78 38 2d 33 0e 54 75 | .a8-3-3|.x8-3.Tu|
|00000060| 74 6f 72 69 61 6c 20 45 | 78 65 72 63 69 73 65 73 |torial E|xercises|
|00000070| 0f 0d 0a 00 20 20 20 20 | 20 10 61 38 2d 33 2d 32 |.... | .a8-3-2|
|00000080| 0e 65 38 2d 33 0e 47 75 | 69 64 65 64 20 45 78 61 |.e8-3.Gu|ided Exa|
|00000090| 6d 70 6c 65 73 0f 0d 0a | 00 0d 0a 00 54 6f 70 69 |mples...|....Topi|
|000000a0| 63 73 20 66 6f 72 20 65 | 78 70 6c 6f 72 61 74 69 |cs for e|xplorati|
|000000b0| 6f 6e 3a 0d 0a 00 0d 0b | 00 20 20 20 20 20 0e 73 |on:.....|. .s|
|000000c0| 38 2d 33 2d 31 0e 56 65 | 63 74 6f 72 20 54 65 72 |8-3-1.Ve|ctor Ter|
|000000d0| 6d 69 6e 6f 6c 6f 67 79 | 0f 0d 0a 00 20 20 20 20 |minology|.... |
|000000e0| 20 0e 73 38 2d 33 2d 32 | 0e 43 6f 6d 70 6f 6e 65 | .s8-3-2|.Compone|
|000000f0| 6e 74 20 46 6f 72 6d 20 | 6f 66 20 61 20 56 65 63 |nt Form |of a Vec|
|00000100| 74 6f 72 0f 0d 0a 00 20 | 20 20 20 20 0e 73 38 2d |tor.... | .s8-|
|00000110| 33 2d 33 0e 56 65 63 74 | 6f 72 20 4f 70 65 72 61 |3-3.Vect|or Opera|
|00000120| 74 69 6f 6e 73 0f 0d 0a | 00 20 20 20 20 20 0e 73 |tions...|. .s|
|00000130| 38 2d 33 2d 34 0e 50 72 | 6f 70 65 72 74 69 65 73 |8-3-4.Pr|operties|
|00000140| 20 6f 66 20 56 65 63 74 | 6f 72 20 41 64 64 69 74 | of Vect|or Addit|
|00000150| 69 6f 6e 20 61 6e 64 20 | 53 63 61 6c 61 72 20 4d |ion and |Scalar M|
|00000160| 75 6c 74 69 70 6c 69 63 | 61 74 69 6f 6e 0f 0d 0a |ultiplic|ation...|
|00000170| 00 20 20 20 20 20 0e 73 | 38 2d 33 2d 35 0e 55 6e |. .s|8-3-5.Un|
|00000180| 69 74 20 56 65 63 74 6f | 72 20 69 6e 20 74 68 65 |it Vecto|r in the|
|00000190| 20 44 69 72 65 63 74 69 | 6f 6e 20 6f 66 20 61 20 | Directi|on of a |
|000001a0| 47 69 76 65 6e 20 56 65 | 63 74 6f 72 0f 0d 0a 00 |Given Ve|ctor....|
|000001b0| 20 20 20 20 20 0e 73 38 | 2d 33 2d 36 0e 53 74 61 | .s8|-3-6.Sta|
|000001c0| 6e 64 61 72 64 20 55 6e | 69 74 20 56 65 63 74 6f |ndard Un|it Vecto|
|000001d0| 72 73 0f 0d 0a 00 20 20 | 20 20 20 0e 73 38 2d 33 |rs.... | .s8-3|
|000001e0| 2d 37 0e 44 69 72 65 63 | 74 69 6f 6e 20 41 6e 67 |-7.Direc|tion Ang|
|000001f0| 6c 65 20 6f 66 20 61 20 | 56 65 63 74 6f 72 0f 0d |le of a |Vector..|
|00000200| 0a 00 53 65 63 74 69 6f | 6e 20 38 2e 33 20 20 56 |..Sectio|n 8.3 V|
|00000210| 65 63 74 6f 72 73 20 69 | 6e 20 74 68 65 20 50 6c |ectors i|n the Pl|
|00000220| 61 6e 65 0d 0b 00 4d 61 | 6e 79 20 61 70 70 6c 69 |ane...Ma|ny appli|
|00000230| 63 61 74 69 6f 6e 73 20 | 69 6e 20 73 63 69 65 6e |cations |in scien|
|00000240| 63 65 20 72 65 71 75 69 | 72 65 20 71 75 61 6e 74 |ce requi|re quant|
|00000250| 69 74 69 65 73 20 74 68 | 61 74 20 68 61 76 65 20 |ities th|at have |
|00000260| 62 6f 74 68 20 61 20 11 | 33 6d 61 67 6e 69 74 75 |both a .|3magnitu|
|00000270| 64 65 20 0d 0a 00 11 31 | 61 6e 64 20 11 33 64 69 |de ....1|and .3di|
|00000280| 72 65 63 74 69 6f 6e 20 | 11 31 61 6e 64 20 63 61 |rection |.1and ca|
|00000290| 6e 6e 6f 74 20 62 65 20 | 63 6f 6d 70 6c 65 74 65 |nnot be |complete|
|000002a0| 6c 79 20 63 68 61 72 61 | 63 74 65 72 69 7a 65 64 |ly chara|cterized|
|000002b0| 20 62 79 20 61 20 73 69 | 6e 67 6c 65 20 72 65 61 | by a si|ngle rea|
|000002c0| 6c 20 6e 75 6d 62 65 72 | 2e 20 20 0d 0a 00 54 6f |l number|. ...To|
|000002d0| 20 64 65 61 6c 20 77 69 | 74 68 20 74 68 65 73 65 | deal wi|th these|
|000002e0| 20 71 75 61 6e 74 69 74 | 69 65 73 2c 20 77 65 20 | quantit|ies, we |
|000002f0| 75 73 65 20 61 20 12 31 | 64 69 72 65 63 74 65 64 |use a .1|directed|
|00000300| 20 6c 69 6e 65 20 73 65 | 67 6d 65 6e 74 12 30 2e | line se|gment.0.|
|00000310| 20 20 41 20 64 69 72 65 | 63 74 65 64 20 0d 0a 00 | A dire|cted ...|
|00000320| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 34 32 | | .42|
|00000330| 33 0d 0b 00 11 31 6c 69 | 6e 65 20 73 65 67 6d 65 |3....1li|ne segme|
|00000340| 6e 74 20 11 33 50 51 20 | 11 31 68 61 73 20 12 31 |nt .3PQ |.1has .1|
|00000350| 69 6e 69 74 69 61 6c 20 | 70 6f 69 6e 74 12 30 20 |initial |point.0 |
|00000360| 11 33 50 20 11 31 61 6e | 64 20 12 31 74 65 72 6d |.3P .1an|d .1term|
|00000370| 69 6e 61 6c 20 70 6f 69 | 6e 74 12 30 20 11 33 51 |inal poi|nt.0 .3Q|
|00000380| 20 11 31 61 6e 64 20 77 | 65 20 64 65 6e 6f 74 65 | .1and w|e denote|
|00000390| 20 69 74 73 0d 0a 00 6c | 65 6e 67 74 68 20 62 79 | its...l|ength by|
|000003a0| 20 11 34 7c 11 33 50 51 | 11 34 7c 11 31 2e 20 20 | .4|.3PQ|.4|.1. |
|000003b0| 54 77 6f 20 64 69 72 65 | 63 74 65 64 20 6c 69 6e |Two dire|cted lin|
|000003c0| 65 20 73 65 67 6d 65 6e | 74 73 20 74 68 61 74 20 |e segmen|ts that |
|000003d0| 68 61 76 65 20 74 68 65 | 20 73 61 6d 65 20 6c 65 |have the| same le|
|000003e0| 6e 67 74 68 20 28 6f 72 | 20 0d 0a 00 6d 61 67 6e |ngth (or| ...magn|
|000003f0| 69 74 75 64 65 29 20 61 | 6e 64 20 64 69 72 65 63 |itude) a|nd direc|
|00000400| 74 69 6f 6e 20 61 72 65 | 20 12 31 65 71 75 69 76 |tion are| .1equiv|
|00000410| 61 6c 65 6e 74 12 30 2e | 20 20 4e 6f 74 65 20 74 |alent.0.| Note t|
|00000420| 68 61 74 20 74 68 65 20 | 69 6e 69 74 69 61 6c 20 |hat the |initial |
|00000430| 61 6e 64 20 74 65 72 6d | 69 6e 61 6c 0d 0a 00 70 |and term|inal...p|
|00000440| 6f 69 6e 74 73 20 64 6f | 20 11 33 6e 6f 74 20 11 |oints do| .3not .|
|00000450| 31 68 61 76 65 20 74 6f | 20 62 65 20 74 68 65 20 |1have to| be the |
|00000460| 73 61 6d 65 2e 20 20 54 | 68 65 20 73 65 74 20 6f |same. T|he set o|
|00000470| 66 20 61 6c 6c 20 64 69 | 72 65 63 74 65 64 20 6c |f all di|rected l|
|00000480| 69 6e 65 20 73 65 67 6d | 65 6e 74 73 20 74 68 61 |ine segm|ents tha|
|00000490| 74 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |t... | |
|000004a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000004b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000004c0| 20 20 20 20 11 34 32 33 | 0d 0b 00 11 31 61 72 65 | .423|....1are|
|000004d0| 20 65 71 75 69 76 61 6c | 65 6e 74 20 74 6f 20 61 | equival|ent to a|
|000004e0| 20 67 69 76 65 6e 20 64 | 69 72 65 63 74 65 64 20 | given d|irected |
|000004f0| 6c 69 6e 65 20 73 65 67 | 6d 65 6e 74 20 11 33 50 |line seg|ment .3P|
|00000500| 51 20 11 31 69 73 20 61 | 20 12 31 76 65 63 74 6f |Q .1is a| .1vecto|
|00000510| 72 20 11 25 76 20 11 31 | 69 6e 20 74 68 65 12 30 |r .%v .1|in the.0|
|00000520| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |... | |
|00000530| 20 20 20 20 20 20 20 20 | 20 20 20 11 34 32 33 0d | | .423.|
|00000540| 0b 00 11 31 12 31 70 6c | 61 6e 65 12 30 2c 20 61 |...1.1pl|ane.0, a|
|00000550| 6e 64 20 77 65 20 77 72 | 69 74 65 20 11 25 76 20 |nd we wr|ite .%v |
|00000560| 11 31 3d 20 11 33 50 51 | 11 31 2e 20 20 28 4c 6f |.1= .3PQ|.1. (Lo|
|00000570| 77 65 72 63 61 73 65 2c | 20 62 6f 6c 64 20 6c 65 |wercase,| bold le|
|00000580| 74 74 65 72 73 20 73 75 | 63 68 20 61 73 20 11 25 |tters su|ch as .%|
|00000590| 75 11 31 2c 20 11 25 76 | 11 31 2c 20 61 6e 64 20 |u.1, .%v|.1, and |
|000005a0| 0d 0a 00 11 25 77 20 11 | 31 61 72 65 20 75 73 75 |....%w .|1are usu|
|000005b0| 61 6c 6c 79 20 75 73 65 | 64 20 74 6f 20 72 65 70 |ally use|d to rep|
|000005c0| 72 65 73 65 6e 74 20 76 | 65 63 74 6f 72 73 2e 29 |resent v|ectors.)|
|000005d0| 0d 0a 00 0d 0b 00 54 68 | 65 20 64 69 72 65 63 74 |......Th|e direct|
|000005e0| 65 64 20 6c 69 6e 65 20 | 73 65 67 6d 65 6e 74 20 |ed line |segment |
|000005f0| 77 69 74 68 20 69 6e 69 | 74 69 61 6c 20 70 6f 69 |with ini|tial poi|
|00000600| 6e 74 20 61 74 20 74 68 | 65 20 6f 72 69 67 69 6e |nt at th|e origin|
|00000610| 20 69 73 20 73 61 69 64 | 20 74 6f 20 62 65 20 69 | is said| to be i|
|00000620| 6e 20 0d 0a 00 12 31 73 | 74 61 6e 64 61 72 64 20 |n ....1s|tandard |
|00000630| 70 6f 73 69 74 69 6f 6e | 12 30 2e 20 20 54 68 69 |position|.0. Thi|
|00000640| 73 20 72 65 70 72 65 73 | 65 6e 74 61 74 69 76 65 |s repres|entative|
|00000650| 20 6f 66 20 74 68 65 20 | 76 65 63 74 6f 72 20 69 | of the |vector i|
|00000660| 73 20 6f 66 74 65 6e 20 | 74 68 65 20 6d 6f 73 74 |s often |the most|
|00000670| 20 0d 0a 00 63 6f 6e 76 | 65 6e 69 65 6e 74 20 6f | ...conv|enient o|
|00000680| 6e 65 20 74 6f 20 77 6f | 72 6b 20 77 69 74 68 2c |ne to wo|rk with,|
|00000690| 20 73 69 6e 63 65 20 69 | 74 20 63 61 6e 20 62 65 | since i|t can be|
|000006a0| 20 75 6e 69 71 75 65 6c | 79 20 72 65 70 72 65 73 | uniquel|y repres|
|000006b0| 65 6e 74 65 64 20 62 79 | 20 74 68 65 20 63 6f 6f |ented by| the coo|
|000006c0| 72 2d 0d 0a 00 64 69 6e | 61 74 65 73 20 6f 66 20 |r-...din|ates of |
|000006d0| 69 74 73 20 74 65 72 6d | 69 6e 61 6c 20 70 6f 69 |its term|inal poi|
|000006e0| 6e 74 2e 20 20 57 65 20 | 63 61 6c 6c 20 74 68 69 |nt. We |call thi|
|000006f0| 73 20 74 68 65 20 12 31 | 63 6f 6d 70 6f 6e 65 6e |s the .1|componen|
|00000700| 74 20 66 6f 72 6d 20 6f | 66 20 61 20 76 65 63 74 |t form o|f a vect|
|00000710| 6f 72 12 30 20 0d 0a 00 | 61 6e 64 20 77 65 20 75 |or.0 ...|and we u|
|00000720| 73 65 20 74 68 65 20 6e | 6f 74 61 74 69 6f 6e 20 |se the n|otation |
|00000730| 11 25 76 20 11 31 3d 20 | 11 34 6b 11 33 76 20 11 |.%v .1= |.4k.3v .|
|00000740| 31 2c 20 11 33 76 20 11 | 34 4b 11 31 2e 20 20 54 |1, .3v .|4K.1. T|
|00000750| 68 65 20 63 6f 6f 72 64 | 69 6e 61 74 65 73 20 11 |he coord|inates .|
|00000760| 33 76 20 20 11 31 61 6e | 64 20 11 33 76 20 20 11 |3v .1an|d .3v .|
|00000770| 31 61 72 65 20 63 61 6c | 6c 65 64 20 0d 0b 00 20 |1are cal|led ... |
|00000780| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|00000790| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 31 | | .21|
|000007a0| 20 20 20 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | 2 | |
|000007b0| 20 20 20 20 20 20 20 20 | 20 31 20 20 20 20 20 20 | | 1 |
|000007c0| 32 0d 0a 00 11 31 74 68 | 65 20 63 6f 6d 70 6f 6e |2....1th|e compon|
|000007d0| 65 6e 74 73 20 6f 66 20 | 11 25 76 11 31 2e 20 20 |ents of |.%v.1. |
|000007e0| 54 68 65 20 12 31 7a 65 | 72 6f 20 76 65 63 74 6f |The .1ze|ro vecto|
|000007f0| 72 12 30 20 69 73 20 64 | 65 6e 6f 74 65 64 20 62 |r.0 is d|enoted b|
|00000800| 79 20 11 25 30 20 11 31 | 3d 20 11 34 6b 11 31 30 |y .%0 .1|= .4k.10|
|00000810| 2c 20 30 11 34 4b 11 31 | 2e 0d 0a 00 53 65 63 74 |, 0.4K.1|....Sect|
|00000820| 69 6f 6e 20 38 2e 33 20 | 20 56 65 63 74 6f 72 73 |ion 8.3 | Vectors|
|00000830| 20 69 6e 20 74 68 65 20 | 50 6c 61 6e 65 0d 0b 00 | in the |Plane...|
|00000840| 54 68 65 20 12 31 63 6f | 6d 70 6f 6e 65 6e 74 20 |The .1co|mponent |
|00000850| 66 6f 72 6d 20 6f 66 20 | 74 68 65 20 76 65 63 74 |form of |the vect|
|00000860| 6f 72 12 30 20 77 69 74 | 68 20 69 6e 69 74 69 61 |or.0 wit|h initia|
|00000870| 6c 20 70 6f 69 6e 74 20 | 11 33 50 20 11 31 3d 20 |l point |.3P .1= |
|00000880| 11 34 6b 11 33 70 20 11 | 31 2c 20 11 33 70 20 11 |.4k.3p .|1, .3p .|
|00000890| 34 4b 20 11 31 61 6e 64 | 20 74 65 72 6d 69 6e 61 |4K .1and| termina|
|000008a0| 6c 20 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |l ... | |
|000008b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000008c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000008d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 | | .|
|000008e0| 32 31 20 20 20 32 0d 0a | 00 11 31 70 6f 69 6e 74 |21 2..|..1point|
|000008f0| 20 11 33 51 20 11 31 3d | 20 11 34 6b 11 33 71 20 | .3Q .1=| .4k.3q |
|00000900| 2c 20 71 20 11 34 4b 20 | 11 31 69 73 20 0d 0b 00 |, q .4K |.1is ...|
|00000910| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 31 20 | | .21 |
|00000920| 20 20 32 0d 0a 00 20 20 | 20 20 20 11 34 32 33 0d | 2... | .423.|
|00000930| 0b 00 20 20 20 20 20 11 | 33 50 51 20 3d 20 11 34 |.. .|3PQ = .4|
|00000940| 6b 11 33 71 20 20 2d 20 | 70 20 2c 20 71 20 20 2d |k.3q - |p , q -|
|00000950| 20 70 20 11 34 4b 20 11 | 33 3d 20 11 34 6b 11 33 | p .4K .|3= .4k.3|
|00000960| 76 20 2c 20 76 20 11 34 | 4b 20 11 33 3d 20 11 25 |v , v .4|K .3= .%|
|00000970| 76 11 33 2e 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |v.3.... | |
|00000980| 20 20 20 11 32 31 20 20 | 20 20 31 20 20 20 32 20 | .21 | 1 2 |
|00000990| 20 20 20 32 20 20 20 20 | 20 20 31 20 20 20 32 0d | 2 | 1 2.|
|000009a0| 0a 00 0d 0b 00 11 31 54 | 68 65 20 12 31 6c 65 6e |......1T|he .1len|
|000009b0| 67 74 68 12 30 20 28 6f | 72 20 6d 61 67 6e 69 74 |gth.0 (o|r magnit|
|000009c0| 75 64 65 29 20 6f 66 20 | 11 25 76 20 11 31 69 73 |ude) of |.%v .1is|
|000009d0| 20 67 69 76 65 6e 20 62 | 79 20 0d 0a 00 20 20 20 | given b|y ... |
|000009e0| 20 20 20 20 20 20 20 20 | 20 20 11 34 67 32 32 32 | | .4g222|
|000009f0| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|00000a00| 32 32 32 20 20 20 20 20 | 67 32 32 32 32 32 32 32 |222 |g2222222|
|00000a10| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |2... | |
|00000a20| 66 20 20 20 20 20 20 20 | 20 20 11 32 32 20 20 20 |f | .22 |
|00000a30| 20 20 20 20 20 20 20 20 | 20 32 20 20 20 20 11 34 | | 2 .4|
|00000a40| 66 20 20 11 32 32 20 20 | 20 20 20 32 0d 0b 00 20 |f .22 | 2... |
|00000a50| 20 20 20 20 11 34 7c 11 | 25 76 11 34 7c 20 11 31 | .4|.|%v.4| .1|
|00000a60| 3d 20 11 34 76 20 11 33 | 28 71 20 20 2d 20 70 20 |= .4v .3|(q - p |
|00000a70| 29 20 20 2b 20 28 71 20 | 20 2d 20 70 20 29 20 20 |) + (q | - p ) |
|00000a80| 3d 20 11 34 76 20 11 33 | 76 20 20 20 2b 20 76 20 |= .4v .3|v + v |
|00000a90| 20 2e 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 | .... | |
|00000aa0| 20 20 20 20 11 32 31 20 | 20 20 20 31 20 20 20 20 | .21 | 1 |
|00000ab0| 20 20 20 32 20 20 20 20 | 32 20 20 20 20 20 20 20 | 2 |2 |
|00000ac0| 20 31 20 20 20 20 20 32 | 0d 0a 00 11 31 49 66 20 | 1 2|....1If |
|00000ad0| 11 34 7c 11 25 76 11 34 | 7c 20 11 31 3d 20 31 2c |.4|.%v.4|| .1= 1,|
|00000ae0| 20 74 68 65 6e 20 11 25 | 76 20 11 31 69 73 20 63 | then .%|v .1is c|
|00000af0| 61 6c 6c 65 64 20 61 20 | 12 31 75 6e 69 74 20 76 |alled a |.1unit v|
|00000b00| 65 63 74 6f 72 12 30 2e | 20 20 4d 6f 72 65 6f 76 |ector.0.| Moreov|
|00000b10| 65 72 2c 20 11 34 7c 11 | 25 76 11 34 7c 20 11 31 |er, .4|.|%v.4| .1|
|00000b20| 3d 20 30 20 69 66 20 61 | 6e 64 20 6f 6e 6c 79 20 |= 0 if a|nd only |
|00000b30| 0d 0a 00 69 66 20 11 25 | 76 20 11 31 69 73 20 74 |...if .%|v .1is t|
|00000b40| 68 65 20 7a 65 72 6f 20 | 76 65 63 74 6f 72 20 11 |he zero |vector .|
|00000b50| 25 30 11 31 2e 0d 0a 00 | 53 65 63 74 69 6f 6e 20 |%0.1....|Section |
|00000b60| 38 2e 33 20 20 56 65 63 | 74 6f 72 73 20 69 6e 20 |8.3 Vec|tors in |
|00000b70| 74 68 65 20 50 6c 61 6e | 65 4c 65 74 20 11 25 75 |the Plan|eLet .%u|
|00000b80| 20 11 31 3d 20 11 34 6b | 11 33 75 20 2c 20 75 20 | .1= .4k|.3u , u |
|00000b90| 11 34 4b 20 11 31 61 6e | 64 20 11 25 76 20 11 31 |.4K .1an|d .%v .1|
|00000ba0| 3d 20 11 34 6b 11 33 76 | 20 2c 20 76 20 11 34 4b |= .4k.3v| , v .4K|
|00000bb0| 20 11 31 62 65 20 76 65 | 63 74 6f 72 73 20 61 6e | .1be ve|ctors an|
|00000bc0| 64 20 6c 65 74 20 11 33 | 6b 20 11 31 62 65 20 61 |d let .3|k .1be a|
|00000bd0| 20 73 63 61 6c 61 72 20 | 28 61 20 72 65 61 6c 20 | scalar |(a real |
|00000be0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 11 32 31 |... | .21|
|00000bf0| 20 20 20 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | 2 | |
|00000c00| 31 20 20 20 32 0d 0a 00 | 11 31 6e 75 6d 62 65 72 |1 2...|.1number|
|00000c10| 29 2e 20 20 54 68 65 6e | 20 74 68 65 20 12 31 73 |). Then| the .1s|
|00000c20| 75 6d 12 30 20 6f 66 20 | 11 25 75 20 11 31 61 6e |um.0 of |.%u .1an|
|00000c30| 64 20 11 25 76 20 11 31 | 69 73 20 74 68 65 20 76 |d .%v .1|is the v|
|00000c40| 65 63 74 6f 72 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |ector...|... |
|00000c50| 11 25 75 20 11 31 2b 20 | 11 25 76 20 11 31 3d 20 |.%u .1+ |.%v .1= |
|00000c60| 11 34 6b 11 33 75 20 20 | 2b 20 76 20 2c 20 75 20 |.4k.3u |+ v , u |
|00000c70| 20 2b 20 76 20 11 34 4b | 0d 0b 00 20 20 20 20 20 | + v .4K|... |
|00000c80| 20 20 20 20 20 20 20 20 | 20 20 11 32 31 20 20 20 | | .21 |
|00000c90| 20 31 20 20 20 32 20 20 | 20 20 32 0d 0a 00 11 31 | 1 2 | 2....1|
|00000ca0| 61 6e 64 20 74 68 65 20 | 12 31 73 63 61 6c 61 72 |and the |.1scalar|
|00000cb0| 20 6d 75 6c 74 69 70 6c | 65 12 30 20 6f 66 20 11 | multipl|e.0 of .|
|00000cc0| 33 6b 20 11 31 74 69 6d | 65 73 20 11 25 75 20 11 |3k .1tim|es .%u .|
|00000cd0| 31 69 73 20 74 68 65 20 | 76 65 63 74 6f 72 0d 0a |1is the |vector..|
|00000ce0| 00 0d 0b 00 20 20 20 20 | 20 11 33 6b 11 25 75 20 |.... | .3k.%u |
|00000cf0| 11 31 3d 20 11 33 6b 11 | 34 6b 11 33 75 20 2c 20 |.1= .3k.|4k.3u , |
|00000d00| 75 20 11 34 4b 20 11 31 | 3d 20 11 34 6b 11 33 6b |u .4K .1|= .4k.3k|
|00000d10| 75 20 2c 20 6b 75 20 11 | 34 4b 11 33 2e 0d 0b 00 |u , ku .|4K.3....|
|00000d20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 31 | | .21|
|00000d30| 20 20 20 32 20 20 20 20 | 20 20 20 31 20 20 20 20 | 2 | 1 |
|00000d40| 32 0d 0a 00 11 31 47 65 | 6f 6d 65 74 72 69 63 61 |2....1Ge|ometrica|
|00000d50| 6c 6c 79 2c 20 77 65 20 | 63 61 6e 20 69 6e 74 65 |lly, we |can inte|
|00000d60| 72 70 72 65 74 20 12 31 | 76 65 63 74 6f 72 20 61 |rpret .1|vector a|
|00000d70| 64 64 69 74 69 6f 6e 12 | 30 20 61 73 20 70 6c 61 |ddition.|0 as pla|
|00000d80| 63 69 6e 67 20 74 68 65 | 20 69 6e 69 74 69 61 6c |cing the| initial|
|00000d90| 20 70 6f 69 6e 74 0d 0a | 00 6f 66 20 11 25 76 20 | point..|.of .%v |
|00000da0| 11 31 61 74 20 74 68 65 | 20 74 65 72 6d 69 6e 61 |.1at the| termina|
|00000db0| 6c 20 70 6f 69 6e 74 20 | 6f 66 20 11 25 75 11 31 |l point |of .%u.1|
|00000dc0| 2e 20 20 53 69 6e 63 65 | 20 74 68 65 20 76 65 63 |. Since| the vec|
|00000dd0| 74 6f 72 20 11 25 75 20 | 11 31 2b 20 11 25 76 20 |tor .%u |.1+ .%v |
|00000de0| 11 31 69 73 20 74 68 65 | 20 64 69 61 67 6f 6e 61 |.1is the| diagona|
|00000df0| 6c 20 6f 66 20 61 0d 0a | 00 70 61 72 61 6c 6c 65 |l of a..|.paralle|
|00000e00| 6c 6f 67 72 61 6d 20 68 | 61 76 69 6e 67 20 11 25 |logram h|aving .%|
|00000e10| 75 20 11 31 61 6e 64 20 | 11 25 76 20 11 31 61 73 |u .1and |.%v .1as|
|00000e20| 20 69 74 73 20 61 64 6a | 61 63 65 6e 74 20 73 69 | its adj|acent si|
|00000e30| 64 65 73 2c 20 77 65 20 | 63 61 6c 6c 20 74 68 69 |des, we |call thi|
|00000e40| 73 20 74 68 65 20 0d 0a | 00 12 31 70 61 72 61 6c |s the ..|..1paral|
|00000e50| 6c 65 6c 6f 67 72 61 6d | 20 6c 61 77 12 30 20 66 |lelogram| law.0 f|
|00000e60| 6f 72 20 76 65 63 74 6f | 72 20 61 64 64 69 74 69 |or vecto|r additi|
|00000e70| 6f 6e 2e 20 20 53 63 61 | 6c 61 72 20 6d 75 6c 74 |on. Sca|lar mult|
|00000e80| 69 70 6c 69 63 61 74 69 | 6f 6e 20 68 61 73 20 61 |iplicati|on has a|
|00000e90| 6e 20 65 71 75 61 6c 6c | 79 0d 0a 00 75 73 65 66 |n equall|y...usef|
|00000ea0| 75 6c 20 67 65 6f 6d 65 | 74 72 69 63 61 6c 20 69 |ul geome|trical i|
|00000eb0| 6e 74 65 72 70 72 65 74 | 61 74 69 6f 6e 2e 20 20 |nterpret|ation. |
|00000ec0| 54 68 65 20 73 63 61 6c | 61 72 20 11 33 6b 20 11 |The scal|ar .3k .|
|00000ed0| 31 72 65 70 72 65 73 65 | 6e 74 73 20 74 68 65 20 |1represe|nts the |
|00000ee0| 66 61 63 74 6f 72 20 62 | 79 20 0d 0a 00 77 68 69 |factor b|y ...whi|
|00000ef0| 63 68 20 11 25 75 20 11 | 31 69 73 20 73 74 72 65 |ch .%u .|1is stre|
|00000f00| 74 63 68 65 64 20 6f 72 | 20 73 68 72 75 6e 6b 2e |tched or| shrunk.|
|00000f10| 20 20 49 66 20 11 33 6b | 20 11 31 3e 20 30 20 74 | If .3k| .1> 0 t|
|00000f20| 68 65 6e 20 74 68 65 20 | 64 69 72 65 63 74 69 6f |hen the |directio|
|00000f30| 6e 20 6f 66 20 11 33 6b | 11 25 75 20 11 31 69 73 |n of .3k|.%u .1is|
|00000f40| 20 74 68 65 20 73 61 6d | 65 0d 0a 00 61 73 20 11 | the sam|e...as .|
|00000f50| 25 75 11 31 2e 20 20 49 | 66 20 11 33 6b 20 11 31 |%u.1. I|f .3k .1|
|00000f60| 3c 20 30 20 74 68 65 6e | 20 74 68 65 20 64 69 72 |< 0 then| the dir|
|00000f70| 65 63 74 69 6f 6e 20 69 | 73 20 6f 70 70 6f 73 69 |ection i|s opposi|
|00000f80| 74 65 2e 0d 0a 00 54 68 | 65 20 12 31 6e 65 67 61 |te....Th|e .1nega|
|00000f90| 74 69 76 65 12 30 20 6f | 66 20 11 25 76 20 11 31 |tive.0 o|f .%v .1|
|00000fa0| 69 73 20 2d 11 25 76 20 | 11 31 3d 20 28 2d 31 29 |is -.%v |.1= (-1)|
|00000fb0| 11 25 76 20 11 31 3d 20 | 11 34 6b 11 33 2d 76 20 |.%v .1= |.4k.3-v |
|00000fc0| 2c 20 2d 76 20 11 34 4b | 20 11 31 61 6e 64 20 74 |, -v .4K| .1and t|
|00000fd0| 68 65 20 12 31 64 69 66 | 66 65 72 65 6e 63 65 12 |he .1dif|ference.|
|00000fe0| 30 20 6f 66 20 11 25 75 | 20 11 31 61 6e 64 20 11 |0 of .%u| .1and .|
|00000ff0| 25 76 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |%v... | |
|00001000| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|00001010| 20 20 20 20 20 20 20 20 | 20 20 11 32 31 20 20 20 | | .21 |
|00001020| 20 32 0d 0a 00 11 31 69 | 73 20 11 25 75 20 11 31 | 2....1i|s .%u .1|
|00001030| 2d 20 11 25 76 20 11 31 | 3d 20 11 25 75 20 11 31 |- .%v .1|= .%u .1|
|00001040| 2b 20 28 2d 11 25 76 11 | 31 29 20 3d 20 11 34 6b |+ (-.%v.|1) = .4k|
|00001050| 11 33 75 20 20 2d 20 76 | 20 2c 20 75 20 20 2d 20 |.3u - v| , u - |
|00001060| 76 20 11 34 4b 11 31 2e | 20 20 47 65 6f 6d 65 74 |v .4K.1.| Geomet|
|00001070| 72 69 63 61 6c 6c 79 2c | 20 11 25 75 20 11 31 2d |rically,| .%u .1-|
|00001080| 20 11 25 76 20 11 31 69 | 73 20 64 72 61 77 6e 20 | .%v .1i|s drawn |
|00001090| 66 72 6f 6d 20 0d 0b 00 | 20 20 20 20 20 20 20 20 |from ...| |
|000010a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000010b0| 11 32 31 20 20 20 20 31 | 20 20 20 32 20 20 20 20 |.21 1| 2 |
|000010c0| 32 0d 0a 00 11 25 76 11 | 31 27 73 20 74 65 72 6d |2....%v.|1's term|
|000010d0| 69 6e 61 6c 20 70 6f 69 | 6e 74 20 74 6f 20 11 25 |inal poi|nt to .%|
|000010e0| 75 11 31 27 73 20 74 65 | 72 6d 69 6e 61 6c 20 70 |u.1's te|rminal p|
|000010f0| 6f 69 6e 74 20 28 11 25 | 75 20 11 31 61 6e 64 20 |oint (.%|u .1and |
|00001100| 11 25 76 20 11 31 69 6e | 20 73 74 61 6e 64 61 72 |.%v .1in| standar|
|00001110| 64 20 70 6f 73 69 74 69 | 6f 6e 29 2e 0d 0a 00 53 |d positi|on)....S|
|00001120| 65 63 74 69 6f 6e 20 38 | 2e 33 20 20 56 65 63 74 |ection 8|.3 Vect|
|00001130| 6f 72 73 20 69 6e 20 74 | 68 65 20 50 6c 61 6e 65 |ors in t|he Plane|
|00001140| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 12 31 |... | .1|
|00001150| 50 72 6f 70 65 72 74 69 | 65 73 20 6f 66 20 56 65 |Properti|es of Ve|
|00001160| 63 74 6f 72 20 41 64 64 | 69 74 69 6f 6e 20 61 6e |ctor Add|ition an|
|00001170| 64 20 53 63 61 6c 61 72 | 20 4d 75 6c 74 69 70 6c |d Scalar| Multipl|
|00001180| 69 63 61 74 69 6f 6e 12 | 30 0d 0a 00 0d 0b 00 4c |ication.|0......L|
|00001190| 65 74 20 11 25 75 11 31 | 2c 20 11 25 76 11 31 2c |et .%u.1|, .%v.1,|
|000011a0| 20 61 6e 64 20 11 25 77 | 20 11 31 62 65 20 76 65 | and .%w| .1be ve|
|000011b0| 63 74 6f 72 73 20 61 6e | 64 20 6c 65 74 20 11 33 |ctors an|d let .3|
|000011c0| 63 20 11 31 61 6e 64 20 | 11 33 64 20 11 31 62 65 |c .1and |.3d .1be|
|000011d0| 20 73 63 61 6c 61 72 73 | 2e 20 20 54 68 65 6e 20 | scalars|. Then |
|000011e0| 74 68 65 20 66 6f 6c 6c | 6f 77 69 6e 67 20 0d 0a |the foll|owing ..|
|000011f0| 00 12 31 70 72 6f 70 65 | 72 74 69 65 73 12 30 20 |..1prope|rties.0 |
|00001200| 61 72 65 20 74 72 75 65 | 2e 0d 0a 00 0d 0b 00 20 |are true|....... |
|00001210| 31 2e 20 11 25 75 20 11 | 31 2b 20 11 25 76 20 11 |1. .%u .|1+ .%v .|
|00001220| 31 3d 20 11 25 76 20 11 | 31 2b 20 11 25 75 20 20 |1= .%v .|1+ .%u |
|00001230| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 | | .2.|
|00001240| 31 43 6f 6d 6d 75 74 61 | 74 69 76 65 20 50 72 6f |1Commuta|tive Pro|
|00001250| 70 65 72 74 79 20 6f 66 | 20 41 64 64 69 74 69 6f |perty of| Additio|
|00001260| 6e 12 30 0d 0a 00 20 11 | 31 32 2e 20 28 11 25 75 |n.0... .|12. (.%u|
|00001270| 20 11 31 2b 20 11 25 76 | 11 31 29 20 2b 20 11 25 | .1+ .%v|.1) + .%|
|00001280| 77 20 11 31 3d 20 11 25 | 75 20 11 31 2b 20 28 11 |w .1= .%|u .1+ (.|
|00001290| 25 76 20 11 31 2b 20 11 | 25 77 11 31 29 20 20 20 |%v .1+ .|%w.1) |
|000012a0| 11 32 12 31 41 73 73 6f | 63 69 61 74 69 76 65 20 |.2.1Asso|ciative |
|000012b0| 50 72 6f 70 65 72 74 79 | 20 6f 66 20 41 64 64 69 |Property| of Addi|
|000012c0| 74 69 6f 6e 12 30 0d 0a | 00 20 11 31 33 2e 20 11 |tion.0..|. .13. .|
|000012d0| 25 75 20 11 31 2b 20 11 | 25 30 20 11 31 3d 20 11 |%u .1+ .|%0 .1= .|
|000012e0| 25 75 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |%u | |
|000012f0| 20 20 20 20 20 11 32 12 | 31 11 35 30 20 11 32 69 | .2.|1.50 .2i|
|00001300| 73 20 74 68 65 20 41 64 | 64 69 74 69 76 65 20 49 |s the Ad|ditive I|
|00001310| 64 65 6e 74 69 74 79 12 | 30 0d 0a 00 20 11 31 34 |dentity.|0... .14|
|00001320| 2e 20 11 25 75 20 11 31 | 2b 20 28 2d 11 25 75 11 |. .%u .1|+ (-.%u.|
|00001330| 31 29 20 3d 20 11 25 30 | 20 20 20 20 20 20 20 20 |1) = .%0| |
|00001340| 20 20 20 20 20 20 20 20 | 11 32 12 31 2d 11 35 75 | |.2.1-.5u|
|00001350| 20 11 32 69 73 20 74 68 | 65 20 41 64 64 69 74 69 | .2is th|e Additi|
|00001360| 76 65 20 49 6e 76 65 72 | 73 65 20 6f 66 20 11 35 |ve Inver|se of .5|
|00001370| 75 11 32 12 30 0d 0a 00 | 20 11 31 35 2e 20 11 33 |u.2.0...| .15. .3|
|00001380| 63 28 64 11 25 75 11 33 | 29 20 3d 20 28 63 64 29 |c(d.%u.3|) = (cd)|
|00001390| 11 25 75 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.%u | |
|000013a0| 20 20 11 32 12 31 41 73 | 73 6f 63 69 61 74 69 76 | .2.1As|sociativ|
|000013b0| 65 20 50 72 6f 70 65 72 | 74 79 20 6f 66 20 53 63 |e Proper|ty of Sc|
|000013c0| 61 6c 61 72 20 4d 75 6c | 74 69 70 6c 69 63 61 74 |alar Mul|tiplicat|
|000013d0| 69 6f 6e 12 30 0d 0a 00 | 20 11 31 36 2e 20 11 33 |ion.0...| .16. .3|
|000013e0| 28 63 20 2b 20 64 29 11 | 25 75 20 11 33 3d 20 63 |(c + d).|%u .3= c|
|000013f0| 11 25 75 20 11 33 2b 20 | 64 11 25 75 20 20 20 20 |.%u .3+ |d.%u |
|00001400| 20 20 20 20 20 20 11 32 | 12 31 44 69 73 74 72 69 | .2|.1Distri|
|00001410| 62 75 74 69 76 65 20 50 | 72 6f 70 65 72 74 79 12 |butive P|roperty.|
|00001420| 30 0d 0a 00 20 11 31 37 | 2e 20 11 33 63 28 11 25 |0... .17|. .3c(.%|
|00001430| 75 20 11 33 2b 20 11 25 | 76 11 33 29 20 3d 20 63 |u .3+ .%|v.3) = c|
|00001440| 11 25 75 20 11 33 2b 20 | 63 11 25 76 20 20 20 20 |.%u .3+ |c.%v |
|00001450| 20 20 20 20 20 20 11 32 | 12 31 44 69 73 74 72 69 | .2|.1Distri|
|00001460| 62 75 74 69 76 65 20 50 | 72 6f 70 65 72 74 79 12 |butive P|roperty.|
|00001470| 30 0d 0a 00 20 11 31 38 | 2e 20 31 28 11 25 75 11 |0... .18|. 1(.%u.|
|00001480| 31 29 20 3d 20 11 25 75 | 20 20 20 20 20 20 20 20 |1) = .%u| |
|00001490| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 | | .2.1|
|000014a0| 31 20 69 73 20 74 68 65 | 20 53 63 61 6c 61 72 20 |1 is the| Scalar |
|000014b0| 4d 75 6c 74 69 70 6c 69 | 63 61 74 69 76 65 20 49 |Multipli|cative I|
|000014c0| 64 65 6e 74 69 74 79 12 | 30 0d 0a 00 20 20 20 20 |dentity.|0... |
|000014d0| 11 31 30 28 11 25 75 11 | 31 29 20 3d 20 11 25 30 |.10(.%u.|1) = .%0|
|000014e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000014f0| 20 20 20 20 11 32 12 31 | 41 6e 79 20 76 65 63 74 | .2.1|Any vect|
|00001500| 6f 72 20 74 69 6d 65 73 | 20 30 20 79 69 65 6c 64 |or times| 0 yield|
|00001510| 73 20 74 68 65 20 7a 65 | 72 6f 20 76 65 63 74 6f |s the ze|ro vecto|
|00001520| 72 12 30 0d 0a 00 20 11 | 31 39 2e 20 11 34 7c 11 |r.0... .|19. .4|.|
|00001530| 33 63 11 25 76 11 34 7c | 20 11 31 3d 20 7c 11 33 |3c.%v.4|| .1= |.3|
|00001540| 63 11 31 7c 11 34 7c 11 | 25 76 11 34 7c 20 20 20 |c.1|.4|.|%v.4| |
|00001550| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 | | .2.1|
|00001560| 54 68 65 20 6c 65 6e 67 | 74 68 20 6f 66 20 63 11 |The leng|th of c.|
|00001570| 25 76 20 11 32 69 73 20 | 7c 63 7c 20 74 69 6d 65 |%v .2is ||c| time|
|00001580| 73 20 74 68 65 20 6c 65 | 6e 67 74 68 20 6f 66 20 |s the le|ngth of |
|00001590| 11 25 76 11 32 12 30 0d | 0a 00 53 65 63 74 69 6f |.%v.2.0.|..Sectio|
|000015a0| 6e 20 38 2e 33 20 20 56 | 65 63 74 6f 72 73 20 69 |n 8.3 V|ectors i|
|000015b0| 6e 20 74 68 65 20 50 6c | 61 6e 65 0d 0b 00 57 65 |n the Pl|ane...We|
|000015c0| 20 73 6f 6d 65 74 69 6d | 65 73 20 6e 65 65 64 20 | sometim|es need |
|000015d0| 61 20 75 6e 69 74 20 76 | 65 63 74 6f 72 20 69 6e |a unit v|ector in|
|000015e0| 20 74 68 65 20 64 69 72 | 65 63 74 69 6f 6e 20 6f | the dir|ection o|
|000015f0| 66 20 61 20 67 69 76 65 | 6e 20 76 65 63 74 6f 72 |f a give|n vector|
|00001600| 20 11 25 76 11 31 2e 20 | 20 57 65 20 63 61 6e 20 | .%v.1. | We can |
|00001610| 0d 0a 00 63 6f 6e 73 74 | 72 75 63 74 20 74 68 65 |...const|ruct the|
|00001620| 20 75 6e 69 74 20 76 65 | 63 74 6f 72 20 61 73 20 | unit ve|ctor as |
|00001630| 66 6f 6c 6c 6f 77 73 2e | 0d 0a 00 20 20 20 20 20 |follows.|... |
|00001640| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|00001650| 20 20 20 20 20 20 20 20 | 11 34 28 20 20 20 29 0d | |.4( ).|
|00001660| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.. | |
|00001670| 20 20 20 20 20 20 20 20 | 20 20 11 25 76 20 20 20 | | .%v |
|00001680| 20 11 34 21 20 11 31 31 | 20 11 34 21 0d 0b 00 20 | .4! .11| .4!... |
|00001690| 20 20 20 20 11 25 75 20 | 11 31 3d 20 75 6e 69 74 | .%u |.1= unit|
|000016a0| 20 76 65 63 74 6f 72 20 | 3d 20 11 34 32 32 32 20 | vector |= .4222 |
|000016b0| 11 31 3d 20 11 34 21 32 | 32 32 21 11 25 76 0d 0b |.1= .4!2|22!.%v..|
|000016c0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |. | |
|000016d0| 20 20 20 20 20 20 20 20 | 11 34 7c 11 25 76 11 34 | |.4|.%v.4|
|000016e0| 7c 20 20 20 21 7c 11 25 | 76 11 34 7c 21 0d 0b 00 || !|.%|v.4|!...|
|000016f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|00001700| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 39 20 20 | | 9 |
|00001710| 20 30 0d 0a 00 11 31 4e | 6f 74 65 20 74 68 61 74 | 0....1N|ote that|
|00001720| 20 11 25 75 20 11 31 69 | 73 20 61 20 73 63 61 6c | .%u .1i|s a scal|
|00001730| 61 72 20 6d 75 6c 74 69 | 70 6c 65 20 6f 66 20 11 |ar multi|ple of .|
|00001740| 25 76 20 11 31 77 69 74 | 68 20 6c 65 6e 67 74 68 |%v .1wit|h length|
|00001750| 20 31 20 61 6e 64 20 74 | 68 65 20 73 61 6d 65 20 | 1 and t|he same |
|00001760| 64 69 72 65 63 74 69 6f | 6e 20 0d 0a 00 61 73 20 |directio|n ...as |
|00001770| 11 25 76 11 31 2e 20 20 | 57 65 20 63 61 6c 6c 20 |.%v.1. |We call |
|00001780| 11 25 75 20 11 31 61 20 | 12 31 75 6e 69 74 20 76 |.%u .1a |.1unit v|
|00001790| 65 63 74 6f 72 20 69 6e | 20 74 68 65 20 64 69 72 |ector in| the dir|
|000017a0| 65 63 74 69 6f 6e 20 6f | 66 20 11 25 76 11 31 12 |ection o|f .%v.1.|
|000017b0| 30 2e 0d 0a 00 53 65 63 | 74 69 6f 6e 20 38 2e 33 |0....Sec|tion 8.3|
|000017c0| 20 20 56 65 63 74 6f 72 | 73 20 69 6e 20 74 68 65 | Vector|s in the|
|000017d0| 20 50 6c 61 6e 65 0d 0b | 00 54 68 65 20 75 6e 69 | Plane..|.The uni|
|000017e0| 74 20 76 65 63 74 6f 72 | 73 20 11 25 69 20 11 31 |t vector|s .%i .1|
|000017f0| 3d 20 11 34 6b 11 31 31 | 2c 20 30 11 34 4b 20 11 |= .4k.11|, 0.4K .|
|00001800| 31 61 6e 64 20 11 25 6a | 20 11 31 3d 20 11 34 6b |1and .%j| .1= .4k|
|00001810| 11 31 30 2c 20 31 11 34 | 4b 20 11 31 61 72 65 20 |.10, 1.4|K .1are |
|00001820| 63 61 6c 6c 65 64 20 74 | 68 65 20 12 31 73 74 61 |called t|he .1sta|
|00001830| 6e 64 61 72 64 20 75 6e | 69 74 20 0d 0a 00 0d 0b |ndard un|it .....|
|00001840| 00 76 65 63 74 6f 72 73 | 12 30 2e 20 20 54 68 65 |.vectors|.0. The|
|00001850| 73 65 20 76 65 63 74 6f | 72 73 20 63 61 6e 20 62 |se vecto|rs can b|
|00001860| 65 20 75 73 65 64 20 74 | 6f 20 72 65 70 72 65 73 |e used t|o repres|
|00001870| 65 6e 74 20 61 6e 79 20 | 76 65 63 74 6f 72 20 11 |ent any |vector .|
|00001880| 25 76 20 11 31 3d 20 11 | 34 6b 11 33 76 20 2c 20 |%v .1= .|4k.3v , |
|00001890| 76 20 11 34 4b 20 11 31 | 61 73 20 0d 0b 00 20 20 |v .4K .1|as ... |
|000018a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000018b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000018c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000018d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|000018e0| 11 32 31 20 20 20 32 0d | 0a 00 11 31 66 6f 6c 6c |.21 2.|...1foll|
|000018f0| 6f 77 73 2e 0d 0a 00 0d | 0b 00 20 20 20 20 20 11 |ows.....|.. .|
|00001900| 25 76 20 11 31 3d 20 11 | 34 6b 11 33 76 20 2c 20 |%v .1= .|4k.3v , |
|00001910| 76 20 11 34 4b 20 11 33 | 3d 20 76 20 11 34 6b 11 |v .4K .3|= v .4k.|
|00001920| 33 31 2c 20 30 11 34 4b | 20 11 33 2b 20 76 20 11 |31, 0.4K| .3+ v .|
|00001930| 34 6b 11 33 30 2c 20 31 | 11 34 4b 20 11 33 3d 20 |4k.30, 1|.4K .3= |
|00001940| 76 20 11 25 69 20 11 33 | 2b 20 76 20 11 25 6a 0d |v .%i .3|+ v .%j.|
|00001950| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 11 32 31 |.. | .21|
|00001960| 20 20 20 32 20 20 20 20 | 20 31 20 20 20 20 20 20 | 2 | 1 |
|00001970| 20 20 20 20 32 20 20 20 | 20 20 20 20 20 20 20 31 | 2 | 1|
|00001980| 20 20 20 20 20 32 0d 0a | 00 11 31 54 68 65 20 73 | 2..|..1The s|
|00001990| 63 61 6c 61 72 73 20 11 | 33 76 20 20 11 31 61 6e |calars .|3v .1an|
|000019a0| 64 20 11 33 76 20 20 11 | 31 61 72 65 20 63 61 6c |d .3v .|1are cal|
|000019b0| 6c 65 64 20 74 68 65 20 | 12 31 68 6f 72 69 7a 6f |led the |.1horizo|
|000019c0| 6e 74 61 6c 12 30 20 61 | 6e 64 20 12 31 76 65 72 |ntal.0 a|nd .1ver|
|000019d0| 74 69 63 61 6c 20 63 6f | 6d 70 6f 6e 65 6e 74 73 |tical co|mponents|
|000019e0| 12 30 20 6f 66 20 0d 0b | 00 20 20 20 20 20 20 20 |.0 of ..|. |
|000019f0| 20 20 20 20 20 20 11 32 | 31 20 20 20 20 20 20 32 | .2|1 2|
|00001a00| 0d 0a 00 11 25 76 11 31 | 2c 20 72 65 73 70 65 63 |....%v.1|, respec|
|00001a10| 74 69 76 65 6c 79 2e 20 | 20 54 68 65 20 76 65 63 |tively. | The vec|
|00001a20| 74 6f 72 20 73 75 6d 20 | 11 33 76 20 11 25 69 20 |tor sum |.3v .%i |
|00001a30| 11 31 2b 20 11 33 76 20 | 11 25 6a 20 11 31 69 73 |.1+ .3v |.%j .1is|
|00001a40| 20 61 20 12 31 6c 69 6e | 65 61 72 20 63 6f 6d 62 | a .1lin|ear comb|
|00001a50| 69 6e 61 74 69 6f 6e 12 | 30 20 6f 66 20 74 68 65 |ination.|0 of the|
|00001a60| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |... | |
|00001a70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | | |
|00001a80| 20 20 20 20 20 11 32 31 | 20 20 20 20 20 32 0d 0a | .21| 2..|
|00001a90| 00 11 31 76 65 63 74 6f | 72 73 20 11 25 69 20 11 |..1vecto|rs .%i .|
|00001aa0| 31 61 6e 64 20 11 25 6a | 11 31 2e 20 20 41 6e 79 |1and .%j|.1. Any|
|00001ab0| 20 76 65 63 74 6f 72 20 | 69 6e 20 74 68 65 20 70 | vector |in the p|
|00001ac0| 6c 61 6e 65 20 63 61 6e | 20 62 65 20 65 78 70 72 |lane can| be expr|
|00001ad0| 65 73 73 65 64 20 61 73 | 20 61 20 6c 69 6e 65 61 |essed as| a linea|
|00001ae0| 72 20 0d 0a 00 0d 0b 00 | 63 6f 6d 62 69 6e 61 74 |r ......|combinat|
|00001af0| 69 6f 6e 20 6f 66 20 74 | 68 65 20 73 74 61 6e 64 |ion of t|he stand|
|00001b00| 61 72 64 20 75 6e 69 74 | 20 76 65 63 74 6f 72 73 |ard unit| vectors|
|00001b10| 20 11 25 69 20 11 31 61 | 6e 64 20 11 25 6a 11 31 | .%i .1a|nd .%j.1|
|00001b20| 2e 0d 0a 00 53 65 63 74 | 69 6f 6e 20 38 2e 33 20 |....Sect|ion 8.3 |
|00001b30| 20 56 65 63 74 6f 72 73 | 20 69 6e 20 74 68 65 20 | Vectors| in the |
|00001b40| 50 6c 61 6e 65 0d 0b 00 | 49 66 20 11 25 75 20 11 |Plane...|If .%u .|
|00001b50| 31 69 73 20 61 20 11 33 | 75 6e 69 74 20 76 65 63 |1is a .3|unit vec|
|00001b60| 74 6f 72 20 11 31 73 75 | 63 68 20 74 68 61 74 20 |tor .1su|ch that |
|00001b70| 11 34 74 20 11 31 69 73 | 20 74 68 65 20 61 6e 67 |.4t .1is| the ang|
|00001b80| 6c 65 20 28 6d 65 61 73 | 75 72 65 64 20 63 6f 75 |le (meas|ured cou|
|00001b90| 6e 74 65 72 63 6c 6f 63 | 6b 77 69 73 65 29 20 0d |ntercloc|kwise) .|
|00001ba0| 0a 00 66 72 6f 6d 20 74 | 68 65 20 70 6f 73 69 74 |..from t|he posit|
|00001bb0| 69 76 65 20 11 33 78 11 | 31 2d 61 78 69 73 20 74 |ive .3x.|1-axis t|
|00001bc0| 6f 20 11 25 75 11 31 2c | 20 74 68 65 6e 20 74 68 |o .%u.1,| then th|
|00001bd0| 65 20 74 65 72 6d 69 6e | 61 6c 20 70 6f 69 6e 74 |e termin|al point|
|00001be0| 20 6f 66 20 11 25 75 20 | 11 31 6c 69 65 73 20 6f | of .%u |.1lies o|
|00001bf0| 6e 20 74 68 65 20 75 6e | 69 74 20 0d 0a 00 63 69 |n the un|it ...ci|
|00001c00| 72 63 6c 65 20 61 6e 64 | 20 77 65 20 68 61 76 65 |rcle and| we have|
|00001c10| 20 0d 0a 00 0d 0b 00 20 | 20 20 20 20 11 25 75 20 | ...... | .%u |
|00001c20| 11 31 3d 20 11 34 6b 11 | 31 63 6f 73 20 11 34 74 |.1= .4k.|1cos .4t|
|00001c30| 11 31 2c 20 73 69 6e 20 | 11 34 74 4b 20 11 31 3d |.1, sin |.4tK .1=|
|00001c40| 20 28 63 6f 73 20 11 34 | 74 11 31 29 11 25 69 20 | (cos .4|t.1).%i |
|00001c50| 11 31 2b 20 28 73 69 6e | 20 11 34 74 11 31 29 11 |.1+ (sin| .4t.1).|
|00001c60| 25 6a 0d 0a 00 0d 0b 00 | 11 31 61 6e 64 20 11 34 |%j......|.1and .4|
|00001c70| 74 20 11 31 69 73 20 63 | 61 6c 6c 65 64 20 74 68 |t .1is c|alled th|
|00001c80| 65 20 12 31 64 69 72 65 | 63 74 69 6f 6e 20 61 6e |e .1dire|ction an|
|00001c90| 67 6c 65 12 30 20 6f 66 | 20 74 68 65 20 76 65 63 |gle.0 of| the vec|
|00001ca0| 74 6f 72 20 11 25 75 11 | 31 2e 0d 0a 00 0d 0a 00 |tor .%u.|1.......|
|00001cb0| 53 75 70 70 6f 73 65 20 | 74 68 61 74 20 11 25 75 |Suppose |that .%u|
|00001cc0| 20 11 31 69 73 20 61 20 | 75 6e 69 74 20 76 65 63 | .1is a |unit vec|
|00001cd0| 74 6f 72 20 77 69 74 68 | 20 64 69 72 65 63 74 69 |tor with| directi|
|00001ce0| 6f 6e 20 61 6e 67 6c 65 | 20 11 34 74 11 31 2e 20 |on angle| .4t.1. |
|00001cf0| 20 49 66 20 11 25 76 20 | 11 31 69 73 20 61 6e 79 | If .%v |.1is any|
|00001d00| 20 76 65 63 74 6f 72 20 | 0d 0a 00 74 68 61 74 20 | vector |...that |
|00001d10| 6d 61 6b 65 73 20 61 6e | 20 61 6e 67 6c 65 20 11 |makes an| angle .|
|00001d20| 34 74 20 11 31 77 69 74 | 68 20 74 68 65 20 70 6f |4t .1wit|h the po|
|00001d30| 73 69 74 69 76 65 20 11 | 33 78 11 31 2d 61 78 69 |sitive .|3x.1-axi|
|00001d40| 73 2c 20 74 68 65 6e 20 | 69 74 20 68 61 73 20 74 |s, then |it has t|
|00001d50| 68 65 20 73 61 6d 65 20 | 64 69 72 65 63 2d 0d 0a |he same |direc-..|
|00001d60| 00 74 69 6f 6e 20 61 73 | 20 11 25 75 20 11 31 61 |.tion as| .%u .1a|
|00001d70| 6e 64 20 77 65 20 63 61 | 6e 20 77 72 69 74 65 0d |nd we ca|n write.|
|00001d80| 0a 00 0d 0b 00 20 20 20 | 20 20 11 25 76 20 11 31 |..... | .%v .1|
|00001d90| 3d 20 11 34 7c 11 25 76 | 11 34 7c 6b 11 31 63 6f |= .4|.%v|.4|k.1co|
|00001da0| 73 20 11 34 74 11 31 2c | 20 73 69 6e 20 11 34 74 |s .4t.1,| sin .4t|
|00001db0| 4b 20 11 31 3d 20 11 34 | 7c 11 25 76 11 34 7c 11 |K .1= .4||.%v.4|.|
|00001dc0| 31 28 63 6f 73 20 11 34 | 74 11 31 29 11 25 69 20 |1(cos .4|t.1).%i |
|00001dd0| 11 31 2b 20 11 34 7c 11 | 25 76 11 34 7c 11 31 28 |.1+ .4|.|%v.4|.1(|
|00001de0| 73 69 6e 20 11 34 74 11 | 31 29 11 25 6a 11 31 2e |sin .4t.|1).%j.1.|
|00001df0| 0d 0a 00 31 00 00 00 d1 | 01 00 00 4d 21 00 00 10 |...1....|...M!...|
|00001e00| 00 00 00 00 00 00 00 73 | 38 2d 33 00 23 02 00 00 |.......s|8-3.#...|
|00001e10| f9 05 00 00 4d 21 00 00 | 02 02 00 00 00 00 00 00 |....M!..|........|
|00001e20| 73 38 2d 33 2d 31 00 3d | 08 00 00 1b 03 00 00 4d |s8-3-1.=|.......M|
|00001e30| 21 00 00 1c 08 00 00 00 | 00 00 00 73 38 2d 33 2d |!.......|...s8-3-|
|00001e40| 32 00 79 0b 00 00 a6 05 | 00 00 4d 21 00 00 58 0b |2.y.....|..M!..X.|
|00001e50| 00 00 00 00 00 00 73 38 | 2d 33 2d 33 00 40 11 00 |......s8|-3-3.@..|
|00001e60| 00 5a 04 00 00 4d 21 00 | 00 1f 11 00 00 00 00 00 |.Z...M!.|........|
|00001e70| 00 73 38 2d 33 2d 34 00 | bb 15 00 00 fa 01 00 00 |.s8-3-4.|........|
|00001e80| 4d 21 00 00 9a 15 00 00 | 00 00 00 00 73 38 2d 33 |M!......|....s8-3|
|00001e90| 2d 35 00 d6 17 00 00 4e | 03 00 00 4d 21 00 00 b5 |-5.....N|...M!...|
|00001ea0| 17 00 00 00 00 00 00 73 | 38 2d 33 2d 36 00 45 1b |.......s|8-3-6.E.|
|00001eb0| 00 00 ae 02 00 00 4d 21 | 00 00 24 1b 00 00 00 00 |......M!|..$.....|
|00001ec0| 00 00 73 38 2d 33 2d 37 | 00 |..s8-3-7|. |
+--------+-------------------------+-------------------------+--------+--------+
