Interactive Algebra & Trigonometry - A Self Guided Study Companion

home *** CD-ROM | disk | FTP | other *** search
/ Interactive Algebra & Tri…f Guided Study Companion / Interactive Algebra and Trigonometry - A Self-Guided Study Companion.iso / tutor / chap_8 / 8-4-2.tut < prev next >
Unknown | 1997-03-20 | 4.4 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 | d6 10 00 00 bb 00 00 00 |TUTOR 06|........|
|00000010| 53 65 63 74 69 6f 6e 20 | 38 2e 34 20 20 56 65 63 |Section |8.4  Vec|
|00000020| 74 6f 72 73 20 61 6e 64 | 20 44 6f 74 20 50 72 6f |tors and| Dot Pro|
|00000030| 64 75 63 74 73 0d 0a 00 | 0d 0a 00 0e 65 38 2d 34 |ducts...|....e8-4|
|00000040| 2d 31 0e 47 75 69 64 65 | 64 20 45 78 61 6d 70 6c |-1.Guide|d Exampl|
|00000050| 65 20 31 0f 20 20 46 69 | 6e 64 69 6e 67 20 61 20 |e 1.  Fi|nding a |
|00000060| 44 6f 74 20 50 72 6f 64 | 75 63 74 0d 0a 00 0d 0b |Dot Prod|uct.....|
|00000070| 00 0e 65 38 2d 34 2d 32 | 0e 47 75 69 64 65 64 20 |..e8-4-2|.Guided |
|00000080| 45 78 61 6d 70 6c 65 20 | 32 0f 20 20 55 73 69 6e |Example |2.  Usin|
|00000090| 67 20 50 72 6f 70 65 72 | 74 69 65 73 20 6f 66 20 |g Proper|ties of |
|000000a0| 44 6f 74 20 50 72 6f 64 | 75 63 74 73 0d 0a 00 0d |Dot Prod|ucts....|
|000000b0| 0b 00 0e 65 38 2d 34 2d | 33 0e 47 75 69 64 65 64 |...e8-4-|3.Guided|
|000000c0| 20 45 78 61 6d 70 6c 65 | 20 33 0f 20 20 46 69 6e | Example| 3.  Fin|
|000000d0| 64 69 6e 67 20 74 68 65 | 20 41 6e 67 6c 65 20 42 |ding the| Angle B|
|000000e0| 65 74 77 65 65 6e 20 54 | 77 6f 20 56 65 63 74 6f |etween T|wo Vecto|
|000000f0| 72 73 0d 0a 00 0d 0b 00 | 0e 65 38 2d 34 2d 34 0e |rs......|.e8-4-4.|
|00000100| 47 75 69 64 65 64 20 45 | 78 61 6d 70 6c 65 20 34 |Guided E|xample 4|
|00000110| 0f 20 20 46 69 6e 64 69 | 6e 67 20 74 68 65 20 50 |.  Findi|ng the P|
|00000120| 72 6f 6a 65 63 74 69 6f | 6e 20 6f 66 20 11 25 75 |rojectio|n of .%u|
|00000130| 20 11 31 4f 6e 74 6f 20 | 11 25 76 0d 0a 00 0d 0b | .1Onto |.%v.....|
|00000140| 00 11 31 0e 65 38 2d 34 | 2d 35 0e 47 75 69 64 65 |..1.e8-4|-5.Guide|
|00000150| 64 20 45 78 61 6d 70 6c | 65 20 35 0f 20 20 46 69 |d Exampl|e 5.  Fi|
|00000160| 6e 64 69 6e 67 20 57 6f | 72 6b 0d 0a 00 0d 0b 00 |nding Wo|rk......|
|00000170| 0e 69 38 2d 34 2d 31 0e | 49 6e 74 65 67 72 61 74 |.i8-4-1.|Integrat|
|00000180| 65 64 20 45 78 61 6d 70 | 6c 65 20 31 0f 20 20 46 |ed Examp|le 1.  F|
|00000190| 69 6e 64 69 6e 67 20 74 | 68 65 20 49 6e 74 65 72 |inding t|he Inter|
|000001a0| 69 6f 72 20 41 6e 67 6c | 65 73 20 6f 66 20 61 20 |ior Angl|es of a |
|000001b0| 54 72 69 61 6e 67 6c 65 | 0d 0a 00 0d 0a 00 53 65 |Triangle|......Se|
|000001c0| 63 74 69 6f 6e 20 38 2e | 34 20 20 56 65 63 74 6f |ction 8.|4  Vecto|
|000001d0| 72 73 20 61 6e 64 20 44 | 6f 74 20 50 72 6f 64 75 |rs and D|ot Produ|
|000001e0| 63 74 73 0d 0b 00 46 69 | 6e 64 20 11 34 6b 11 31 |cts...Fi|nd .4k.1|
|000001f0| 36 2c 20 37 11 34 4b 20 | 2a 20 6b 11 31 33 2c 20 |6, 7.4K |* k.13, |
|00000200| 34 11 34 4b 11 31 2e 0d | 0a 00 0d 0a 00 13 12 31 |4.4K.1..|.......1|
|00000210| 53 4f 4c 55 54 49 4f 4e | 12 30 0d 0a 00 0d 0a 00 |SOLUTION|.0......|
|00000220| 20 20 20 20 11 34 6b 11 | 31 36 2c 20 37 11 34 4b |    .4k.|16, 7.4K|
|00000230| 20 2a 20 6b 11 31 33 2c | 20 34 11 34 4b 20 11 31 | * k.13,| 4.4K .1|
|00000240| 3d 20 36 28 33 29 20 2b | 20 37 28 34 29 13 0d 0a |= 6(3) +| 7(4)...|
|00000250| 00 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|00000260| 20 20 20 20 20 20 20 20 | 3d 20 31 38 20 2b 20 32 |        |= 18 + 2|
|00000270| 38 13 0d 0a 00 0d 0a 00 | 20 20 20 20 20 20 20 20 |8.......|        |
|00000280| 20 20 20 20 20 20 20 20 | 20 20 20 20 3d 20 34 36 |        |    = 46|
|00000290| 0d 0a 00 0d 0b 00 53 65 | 63 74 69 6f 6e 20 38 2e |......Se|ction 8.|
|000002a0| 34 20 20 56 65 63 74 6f | 72 73 20 61 6e 64 20 44 |4  Vecto|rs and D|
|000002b0| 6f 74 20 50 72 6f 64 75 | 63 74 73 0d 0b 00 4c 65 |ot Produ|cts...Le|
|000002c0| 74 20 11 25 75 20 11 31 | 3d 20 11 34 6b 11 31 2d |t .%u .1|= .4k.1-|
|000002d0| 32 2c 20 35 11 34 4b 11 | 31 2c 20 11 25 76 20 11 |2, 5.4K.|1, .%v .|
|000002e0| 31 3d 20 11 34 6b 11 31 | 33 2c 20 37 11 34 4b 11 |1= .4k.1|3, 7.4K.|
|000002f0| 31 2c 20 61 6e 64 20 11 | 25 77 20 11 31 3d 20 11 |1, and .|%w .1= .|
|00000300| 34 6b 11 31 32 2c 20 32 | 11 34 4b 11 31 2e 20 20 |4k.12, 2|.4K.1.  |
|00000310| 46 69 6e 64 20 28 11 25 | 75 20 11 34 2a 20 11 25 |Find (.%|u .4* .%|
|00000320| 76 11 31 29 11 25 77 11 | 31 2e 0d 0a 00 0d 0a 00 |v.1).%w.|1.......|
|00000330| 13 12 31 53 4f 4c 55 54 | 49 4f 4e 12 30 0d 0a 00 |..1SOLUT|ION.0...|
|00000340| 0d 0a 00 42 65 67 69 6e | 20 62 79 20 66 69 6e 64 |...Begin| by find|
|00000350| 69 6e 67 20 74 68 65 20 | 64 6f 74 20 70 72 6f 64 |ing the |dot prod|
|00000360| 75 63 74 20 6f 66 20 11 | 25 75 20 11 31 61 6e 64 |uct of .|%u .1and|
|00000370| 20 11 25 76 11 31 2e 0d | 0a 00 0d 0a 00 20 20 20 | .%v.1..|.....   |
|00000380| 20 11 25 75 20 11 34 2a | 20 11 25 76 20 11 31 3d | .%u .4*| .%v .1=|
|00000390| 20 11 34 6b 11 31 2d 32 | 2c 20 35 11 34 4b 20 2a | .4k.1-2|, 5.4K *|
|000003a0| 20 6b 11 31 33 2c 20 37 | 11 34 4b 11 31 13 0d 0a | k.13, 7|.4K.1...|
|000003b0| 00 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 3d 20 |....    |      = |
|000003c0| 28 2d 32 29 28 33 29 20 | 2b 20 35 28 37 29 13 0d |(-2)(3) |+ 5(7)..|
|000003d0| 0a 00 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 3d |.....   |       =|
|000003e0| 20 32 39 13 0d 0a 00 0d | 0a 00 4e 6f 77 2c 20 79 | 29.....|..Now, y|
|000003f0| 6f 75 20 63 61 6e 20 66 | 69 6e 64 20 28 11 25 75 |ou can f|ind (.%u|
|00000400| 20 11 34 2a 20 11 25 76 | 11 31 29 11 25 77 11 31 | .4* .%v|.1).%w.1|
|00000410| 2e 0d 0a 00 0d 0a 00 20 | 20 20 20 28 11 25 75 20 |....... |   (.%u |
|00000420| 11 34 2a 20 11 25 76 11 | 31 29 11 25 77 20 11 31 |.4* .%v.|1).%w .1|
|00000430| 3d 20 32 39 11 34 6b 11 | 31 32 2c 20 32 11 34 4b |= 29.4k.|12, 2.4K|
|00000440| 11 31 13 0d 0a 00 0d 0a | 00 20 20 20 20 20 20 20 |.1......|.       |
|00000450| 20 20 20 20 20 20 3d 20 | 11 34 6b 11 31 35 38 2c |      = |.4k.158,|
|00000460| 20 35 38 11 34 4b 11 31 | 0d 0a 00 0d 0a 00 53 65 | 58.4K.1|......Se|
|00000470| 63 74 69 6f 6e 20 38 2e | 34 20 20 56 65 63 74 6f |ction 8.|4  Vecto|
|00000480| 72 73 20 61 6e 64 20 44 | 6f 74 20 50 72 6f 64 75 |rs and D|ot Produ|
|00000490| 63 74 73 0d 0b 00 46 69 | 6e 64 20 74 68 65 20 61 |cts...Fi|nd the a|
|000004a0| 6e 67 6c 65 20 62 65 74 | 77 65 65 6e 20 11 25 75 |ngle bet|ween .%u|
|000004b0| 20 11 31 3d 20 11 34 6b | 11 31 33 2c 20 34 11 34 | .1= .4k|.13, 4.4|
|000004c0| 4b 20 11 31 61 6e 64 20 | 11 25 76 20 11 31 3d 20 |K .1and |.%v .1= |
|000004d0| 11 34 6b 11 31 31 2c 20 | 35 11 34 4b 11 31 2e 0d |.4k.11, |5.4K.1..|
|000004e0| 0a 00 0d 0b 00 13 12 31 | 53 4f 4c 55 54 49 4f 4e |.......1|SOLUTION|
|000004f0| 12 30 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 |.0......|        |
|00000500| 20 20 20 20 20 11 25 75 | 20 11 34 2a 20 11 25 76 |     .%u| .4* .%v|
|00000510| 0d 0b 00 20 20 20 20 11 | 31 63 6f 73 20 11 34 74 |...    .|1cos .4t|
|00000520| 20 11 31 3d 20 11 34 32 | 32 32 32 32 32 32 0d 0b | .1= .42|222222..|
|00000530| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 7c 11 25 |.       |     |.%|
|00000540| 75 11 34 7c 20 7c 11 25 | 76 11 34 7c 20 11 31 13 |u.4| |.%|v.4| .1.|
|00000550| 0d 0a 00 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|00000560| 20 20 20 11 34 6b 11 31 | 33 2c 20 34 11 34 4b 20 |   .4k.1|3, 4.4K |
|00000570| 2a 20 6b 11 31 31 2c 20 | 35 11 34 4b 0d 0b 00 20 |* k.11, |5.4K... |
|00000580| 20 20 20 20 20 20 20 20 | 20 11 31 3d 20 11 34 32 |        | .1= .42|
|00000590| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|000005a0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 7c |...     |       ||
|000005b0| 6b 11 31 33 2c 20 34 11 | 34 4b 7c 20 7c 6b 11 31 |k.13, 4.|4K| |k.1|
|000005c0| 31 2c 20 35 11 34 4b 7c | 11 31 13 0d 0a 00 0d 0a |1, 5.4K||.1......|
|000005d0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 32 |.       |       2|
|000005e0| 33 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 3d 20 |3...    |      = |
|000005f0| 11 34 32 32 32 32 32 0d | 0b 00 20 20 20 20 20 20 |.422222.|..      |
|00000600| 20 20 20 20 20 20 20 20 | 44 32 32 0d 0b 00 20 20 |        |D22...  |
|00000610| 20 20 20 20 20 20 20 20 | 20 20 11 31 35 11 34 53 |        |  .15.4S|
|00000620| 20 11 31 32 36 20 13 0d | 0a 00 0d 0a 00 54 68 69 | .126 ..|.....Thi|
|00000630| 73 20 69 6d 70 6c 69 65 | 73 20 74 68 61 74 20 74 |s implie|s that t|
|00000640| 68 65 20 61 6e 67 6c 65 | 20 62 65 74 77 65 65 6e |he angle| between|
|00000650| 20 74 68 65 20 74 77 6f | 20 76 65 63 74 6f 72 73 | the two| vectors|
|00000660| 20 69 73 0d 0a 00 0d 0a | 00 20 20 20 20 20 20 20 | is.....|.       |
|00000670| 20 20 20 20 20 20 20 20 | 20 20 32 33 20 20 20 20 |        |  23    |
|00000680| 20 20 20 20 11 34 6f 0d | 0b 00 20 20 20 20 74 20 |    .4o.|..    t |
|00000690| 11 31 3d 20 61 72 63 63 | 6f 73 20 11 34 32 32 32 |.1= arcc|os .4222|
|000006a0| 32 32 20 7e 20 11 31 32 | 35 2e 36 20 2e 0d 0b 00 |22 ~ .12|5.6 ....|
|000006b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000006c0| 20 11 34 44 32 32 0d 0b | 00 20 20 20 20 20 20 20 | .4D22..|.       |
|000006d0| 20 20 20 20 20 20 20 20 | 11 31 35 11 34 53 20 11 |        |.15.4S .|
|000006e0| 31 32 36 0d 0a 00 0d 0b | 00 53 65 63 74 69 6f 6e |126.....|.Section|
|000006f0| 20 38 2e 34 20 20 56 65 | 63 74 6f 72 73 20 61 6e | 8.4  Ve|ctors an|
|00000700| 64 20 44 6f 74 20 50 72 | 6f 64 75 63 74 73 0d 0b |d Dot Pr|oducts..|
|00000710| 00 46 69 6e 64 20 74 68 | 65 20 70 72 6f 6a 65 63 |.Find th|e projec|
|00000720| 74 69 6f 6e 20 6f 66 20 | 11 25 75 20 11 31 3d 20 |tion of |.%u .1= |
|00000730| 11 34 6b 11 31 32 2c 20 | 2d 34 11 34 4b 20 11 31 |.4k.12, |-4.4K .1|
|00000740| 6f 6e 74 6f 20 11 25 76 | 20 11 31 3d 20 11 34 6b |onto .%v| .1= .4k|
|00000750| 11 31 35 2c 20 31 11 34 | 4b 11 31 2e 0d 0a 00 0d |.15, 1.4|K.1.....|
|00000760| 0b 00 13 12 31 53 4f 4c | 55 54 49 4f 4e 12 30 0d |....1SOL|UTION.0.|
|00000770| 0a 00 0d 0b 00 54 68 65 | 20 70 72 6f 6a 65 63 74 |.....The| project|
|00000780| 69 6f 6e 20 6f 66 20 11 | 25 75 20 11 31 6f 6e 74 |ion of .|%u .1ont|
|00000790| 6f 20 11 25 76 20 11 31 | 69 73 20 0d 0a 00 0d 0a |o .%v .1|is .....|
|000007a0| 00 20 20 20 20 11 25 77 | 20 20 11 31 3d 20 70 72 |.    .%w|  .1= pr|
|000007b0| 6f 6a 20 11 25 75 0d 0b | 00 20 20 20 20 20 11 32 |oj .%u..|.     .2|
|000007c0| 31 20 20 20 20 20 20 20 | 76 0d 0a 00 0d 0b 00 20 |1       |v...... |
|000007d0| 20 20 20 20 20 20 20 20 | 11 34 28 20 20 20 20 20 |        |.4(     |
|000007e0| 29 0d 0b 00 20 20 20 20 | 20 20 20 20 20 21 11 25 |)...    |     !.%|
|000007f0| 75 20 11 34 2a 20 11 25 | 76 11 34 21 0d 0b 00 20 |u .4* .%|v.4!... |
|00000800| 20 20 20 20 20 20 11 31 | 3d 20 11 34 21 32 32 32 |      .1|= .4!222|
|00000810| 32 32 21 0d 0b 00 20 20 | 20 20 20 20 20 20 20 21 |22!...  |       !|
|00000820| 20 20 20 20 11 32 32 11 | 34 21 0d 0b 00 20 20 20 |    .22.|4!...   |
|00000830| 20 20 20 20 20 20 21 20 | 7c 11 25 76 11 34 7c 20 |      ! ||.%v.4| |
|00000840| 21 0d 0b 00 20 20 20 20 | 20 20 20 20 20 39 20 20 |!...    |     9  |
|00000850| 20 20 20 30 20 20 20 11 | 31 13 0d 0a 00 0d 0b 00 |   0   .|1.......|
|00000860| 20 20 20 20 20 20 20 20 | 20 11 34 28 20 20 29 0d |        | .4(  ).|
|00000870| 0b 00 20 20 20 20 20 20 | 20 20 20 21 20 11 31 36 |..      |   ! .16|
|00000880| 11 34 21 0d 0b 00 20 20 | 20 20 20 20 20 11 31 3d |.4!...  |     .1=|
|00000890| 20 11 34 21 32 32 21 6b | 11 31 35 2c 20 31 11 34 | .4!22!k|.15, 1.4|
|000008a0| 4b 0d 0b 00 20 20 20 20 | 20 20 20 20 20 21 11 31 |K...    |     !.1|
|000008b0| 32 36 11 34 21 0d 0b 00 | 20 20 20 20 20 20 20 20 |26.4!...|        |
|000008c0| 20 39 20 20 30 20 20 20 | 20 20 20 20 11 31 13 0d | 9  0   |    .1..|
|000008d0| 0a 00 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 31 |.....   |       1|
|000008e0| 35 20 20 20 33 0d 0b 00 | 20 20 20 20 20 20 20 3d |5   3...|       =|
|000008f0| 20 11 34 6b 32 32 11 31 | 2c 20 11 34 32 32 4b 11 | .4k22.1|, .422K.|
|00000900| 31 2e 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 31 |1....   |       1|
|00000910| 33 20 20 31 33 0d 0a 00 | 0d 0a 00 53 65 63 74 69 |3  13...|...Secti|
|00000920| 6f 6e 20 38 2e 34 20 20 | 56 65 63 74 6f 72 73 20 |on 8.4  |Vectors |
|00000930| 61 6e 64 20 44 6f 74 20 | 50 72 6f 64 75 63 74 73 |and Dot |Products|
|00000940| 0d 0b 00 41 6e 20 6f 62 | 6a 65 63 74 20 69 73 20 |...An ob|ject is |
|00000950| 70 75 6c 6c 65 64 20 31 | 30 20 66 65 65 74 20 61 |pulled 1|0 feet a|
|00000960| 63 72 6f 73 73 20 61 20 | 66 6c 6f 6f 72 2c 20 75 |cross a |floor, u|
|00000970| 73 69 6e 67 20 61 20 66 | 6f 72 63 65 20 6f 66 20 |sing a f|orce of |
|00000980| 38 35 20 70 6f 75 6e 64 | 73 2e 20 20 46 69 6e 64 |85 pound|s.  Find|
|00000990| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|000009a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000009b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000009c0| 20 20 20 20 11 34 6f 0d | 0b 00 11 31 74 68 65 20 |    .4o.|...1the |
|000009d0| 77 6f 72 6b 20 64 6f 6e | 65 20 69 66 20 74 68 65 |work don|e if the|
|000009e0| 20 64 69 72 65 63 74 69 | 6f 6e 20 6f 66 20 74 68 | directi|on of th|
|000009f0| 65 20 66 6f 72 63 65 20 | 69 73 20 36 30 20 20 61 |e force |is 60  a|
|00000a00| 62 6f 76 65 20 74 68 65 | 20 68 6f 72 69 7a 6f 6e |bove the| horizon|
|00000a10| 74 61 6c 20 28 73 65 65 | 0d 0a 00 0d 0b 00 66 69 |tal (see|......fi|
|00000a20| 67 75 72 65 29 2e 0d 0a | 00 14 38 2d 34 2d 47 45 |gure)...|..8-4-GE|
|00000a30| 35 2e 61 74 14 30 14 37 | 14 34 33 14 39 14 0d 0a |5.at.0.7|.43.9...|
|00000a40| 00 0d 0a 00 0d 0a 00 0d | 0a 00 0d 0a 00 0d 0a 00 |........|........|
|00000a50| 0d 0a 00 0d 0a 00 0d 0b | 00 13 12 31 53 4f 4c 55 |........|...1SOLU|
|00000a60| 54 49 4f 4e 12 30 0d 0a | 00 0d 0a 00 55 73 69 6e |TION.0..|....Usin|
|00000a70| 67 20 61 20 70 72 6f 6a | 65 63 74 69 6f 6e 2c 20 |g a proj|ection, |
|00000a80| 79 6f 75 20 63 61 6e 20 | 63 61 6c 63 75 6c 61 74 |you can |calculat|
|00000a90| 65 20 74 68 65 20 77 6f | 72 6b 20 61 73 20 66 6f |e the wo|rk as fo|
|00000aa0| 6c 6c 6f 77 73 2e 0d 0a | 00 0d 0a 00 0d 0b 00 20 |llows...|....... |
|00000ab0| 20 20 20 11 33 57 20 11 | 31 3d 20 11 34 7c 11 31 |   .3W .|1= .4|.1|
|00000ac0| 70 72 6f 6a 20 20 11 25 | 46 11 34 7c 20 7c 11 33 |proj  .%|F.4| |.3|
|00000ad0| 50 51 11 34 7c 0d 0b 00 | 20 20 20 20 20 20 20 20 |PQ.4|...|        |
|00000ae0| 20 20 20 20 20 11 32 50 | 51 20 20 20 20 20 20 20 |     .2P|Q       |
|00000af0| 20 20 11 31 13 0d 0a 00 | 0d 0a 00 20 20 20 20 20 |  .1....|...     |
|00000b00| 20 20 20 20 20 20 20 20 | 20 20 11 34 6f 0d 0b 00 |        |  .4o...|
|00000b10| 20 20 20 20 20 20 11 31 | 3d 20 28 63 6f 73 20 36 |      .1|= (cos 6|
|00000b20| 30 20 29 11 34 7c 11 25 | 46 11 34 7c 20 7c 11 33 |0 ).4|.%|F.4| |.3|
|00000b30| 50 51 11 34 7c 20 20 11 | 31 13 0d 0a 00 0d 0a 00 |PQ.4|  .|1.......|
|00000b40| 20 20 20 20 20 20 20 20 | 31 0d 0b 00 20 20 20 20 |        |1...    |
|00000b50| 20 20 3d 20 11 34 32 11 | 31 28 38 35 29 28 31 30 |  = .42.|1(85)(10|
|00000b60| 29 0d 0b 00 20 20 20 20 | 20 20 20 20 32 20 20 20 |)...    |    2   |
|00000b70| 20 20 20 20 20 20 13 0d | 0a 00 0d 0a 00 20 20 20 |      ..|.....   |
|00000b80| 20 20 20 3d 20 34 32 35 | 20 66 6f 6f 74 2d 70 6f |   = 425| foot-po|
|00000b90| 75 6e 64 73 20 13 0d 0a | 00 0d 0a 00 54 68 75 73 |unds ...|....Thus|
|00000ba0| 2c 20 74 68 65 20 77 6f | 72 6b 20 64 6f 6e 65 20 |, the wo|rk done |
|00000bb0| 69 73 20 34 32 35 20 66 | 6f 6f 74 2d 70 6f 75 6e |is 425 f|oot-poun|
|00000bc0| 64 73 2e 0d 0a 00 0d 0a | 00 53 65 63 74 69 6f 6e |ds......|.Section|
|00000bd0| 20 38 2e 34 20 20 56 65 | 63 74 6f 72 73 20 61 6e | 8.4  Ve|ctors an|
|00000be0| 64 20 44 6f 74 20 50 72 | 6f 64 75 63 74 73 0d 0b |d Dot Pr|oducts..|
|00000bf0| 00 55 73 65 20 76 65 63 | 74 6f 72 73 20 74 6f 20 |.Use vec|tors to |
|00000c00| 66 69 6e 64 20 74 68 65 | 20 69 6e 74 65 72 69 6f |find the| interio|
|00000c10| 72 20 61 6e 67 6c 65 73 | 20 6f 66 20 61 20 74 72 |r angles| of a tr|
|00000c20| 69 61 6e 67 6c 65 20 77 | 69 74 68 20 76 65 72 74 |iangle w|ith vert|
|00000c30| 69 63 65 73 20 28 32 2c | 20 30 29 2c 0d 0a 00 0d |ices (2,| 0),....|
|00000c40| 0b 00 28 2d 31 2c 20 33 | 29 2c 20 28 2d 33 2c 20 |..(-1, 3|), (-3, |
|00000c50| 31 29 2e 0d 0a 00 0d 0a | 00 13 12 31 53 4f 4c 55 |1)......|...1SOLU|
|00000c60| 54 49 4f 4e 12 30 0d 0a | 00 0d 0a 00 61 29 20 46 |TION.0..|....a) F|
|00000c70| 69 72 73 74 20 6d 6f 76 | 65 20 74 68 65 20 74 72 |irst mov|e the tr|
|00000c80| 69 61 6e 67 6c 65 20 73 | 6f 20 6f 6e 65 20 6f 66 |iangle s|o one of|
|00000c90| 20 74 68 65 20 76 65 72 | 74 69 63 65 73 20 69 73 | the ver|tices is|
|00000ca0| 20 61 74 20 74 68 65 20 | 6f 72 69 67 69 6e 2e 0d | at the |origin..|
|00000cb0| 0a 00 0d 0b 00 20 20 20 | 43 68 6f 6f 73 69 6e 67 |.....   |Choosing|
|00000cc0| 20 28 32 2c 20 30 29 2c | 20 74 68 65 20 76 65 72 | (2, 0),| the ver|
|00000cd0| 74 69 63 65 73 20 62 65 | 63 6f 6d 65 20 28 30 2c |tices be|come (0,|
|00000ce0| 20 30 29 2c 20 28 2d 33 | 2c 20 33 29 2c 20 61 6e | 0), (-3|, 3), an|
|00000cf0| 64 20 28 2d 35 2c 20 31 | 29 2e 13 0d 0a 00 0d 0a |d (-5, 1|).......|
|00000d00| 00 62 29 20 4e 6f 77 20 | 77 65 20 63 61 6e 20 66 |.b) Now |we can f|
|00000d10| 69 6e 64 20 74 68 65 20 | 61 6e 67 6c 65 20 62 65 |ind the |angle be|
|00000d20| 74 77 65 65 6e 20 11 25 | 75 20 11 31 3d 20 11 34 |tween .%|u .1= .4|
|00000d30| 6b 11 31 2d 33 2c 20 33 | 11 34 4b 20 11 31 61 6e |k.1-3, 3|.4K .1an|
|00000d40| 64 20 11 25 76 20 11 31 | 3d 20 11 34 6b 11 31 2d |d .%v .1|= .4k.1-|
|00000d50| 35 2c 20 31 11 34 4b 11 | 31 2e 0d 0a 00 0d 0a 00 |5, 1.4K.|1.......|
|00000d60| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 34 6b 11 |        |    .4k.|
|00000d70| 31 2d 33 2c 20 33 11 34 | 4b 20 2a 20 6b 11 31 2d |1-3, 3.4|K * k.1-|
|00000d80| 35 2c 20 31 11 34 4b 20 | 20 20 20 20 20 20 20 11 |5, 1.4K |       .|
|00000d90| 31 31 38 20 20 20 20 20 | 20 20 33 0d 0b 00 20 20 |118     |  3...  |
|00000da0| 20 63 6f 73 20 11 34 74 | 20 11 31 3d 20 11 34 32 | cos .4t| .1= .42|
|00000db0| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|00000dc0| 32 32 20 11 31 3d 20 11 | 34 32 32 32 32 32 32 32 |22 .1= .|42222222|
|00000dd0| 32 20 11 31 3d 20 11 34 | 32 32 32 32 0d 0b 00 20 |2 .1= .4|2222... |
|00000de0| 20 20 20 20 20 20 20 20 | 20 20 7c 6b 11 31 2d 33 |        |  |k.1-3|
|00000df0| 2c 20 33 11 34 4b 7c 20 | 7c 6b 11 31 2d 35 2c 20 |, 3.4K| ||k.1-5, |
|00000e00| 31 11 34 4b 7c 20 20 20 | 20 44 32 32 20 44 32 32 |1.4K|   | D22 D22|
|00000e10| 20 20 20 20 44 32 32 0d | 0b 00 20 20 20 20 20 20 |    D22.|..      |
|00000e20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e30| 20 20 20 20 20 20 20 20 | 20 20 20 53 20 11 31 31 |        |   S .11|
|00000e40| 38 11 34 53 20 11 31 32 | 36 20 20 20 11 34 53 20 |8.4S .12|6   .4S |
|00000e50| 11 31 31 33 20 13 0d 0a | 00 0d 0a 00 20 20 20 20 |.113 ...|....    |
|00000e60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e70| 33 20 20 20 20 20 20 20 | 20 11 34 6f 0d 0b 00 20 |3       | .4o... |
|00000e80| 20 20 20 20 20 20 74 20 | 11 31 3d 20 61 72 63 63 |      t |.1= arcc|
|00000e90| 6f 73 20 11 34 32 32 32 | 32 20 7e 20 11 31 33 33 |os .4222|2 ~ .133|
|00000ea0| 2e 37 20 2e 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |.7 .... |        |
|00000eb0| 20 20 20 20 20 20 20 20 | 20 20 11 34 44 32 32 0d |        |  .4D22.|
|00000ec0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00000ed0| 20 20 20 20 53 20 11 31 | 31 33 0d 0b 00 20 20 20 |    S .1|13...   |
|00000ee0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000ef0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000f00| 20 20 20 20 20 20 0d 0a | 00 0d 0a 00 63 29 20 53 |      ..|....c) S|
|00000f10| 6c 69 64 69 6e 67 20 74 | 68 65 20 74 72 69 61 6e |liding t|he trian|
|00000f20| 67 6c 65 20 73 6f 20 61 | 6e 6f 74 68 65 72 20 76 |gle so a|nother v|
|00000f30| 65 72 74 65 78 20 69 73 | 20 61 74 20 74 68 65 20 |ertex is| at the |
|00000f40| 6f 72 69 67 69 6e 2c 20 | 74 68 65 20 76 65 72 74 |origin, |the vert|
|00000f50| 69 63 65 73 0d 0a 00 0d | 0b 00 20 20 20 62 65 63 |ices....|..   bec|
|00000f60| 6f 6d 65 20 28 33 2c 20 | 2d 33 29 2c 20 28 30 2c |ome (3, |-3), (0,|
|00000f70| 20 30 29 2c 20 61 6e 64 | 20 28 2d 32 2c 20 2d 32 | 0), and| (-2, -2|
|00000f80| 29 2e 13 0d 0a 00 0d 0a | 00 64 29 20 4c 65 74 20 |).......|.d) Let |
|00000f90| 11 25 75 20 11 31 3d 20 | 11 34 6b 11 31 33 2c 20 |.%u .1= |.4k.13, |
|00000fa0| 2d 33 11 34 4b 20 11 31 | 61 6e 64 20 11 25 76 20 |-3.4K .1|and .%v |
|00000fb0| 11 31 3d 20 11 34 6b 11 | 31 2d 32 2c 20 2d 32 11 |.1= .4k.|1-2, -2.|
|00000fc0| 34 4b 11 31 2e 20 20 54 | 68 65 20 63 72 6f 73 73 |4K.1.  T|he cross|
|00000fd0| 20 70 72 6f 64 75 63 74 | 20 11 25 75 20 11 34 2a | product| .%u .4*|
|00000fe0| 20 11 25 76 20 11 31 3d | 20 11 34 4f 11 31 2c 20 | .%v .1=| .4O.1, |
|00000ff0| 74 68 75 73 20 74 68 65 | 0d 0a 00 20 20 20 20 20 |thus the|...     |
|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 20 20 20 20 20 20 |        |        |
|00001020| 20 20 20 11 34 6f 0d 0b | 00 20 20 20 11 31 61 6e |   .4o..|.   .1an|
|00001030| 67 6c 65 20 62 65 74 77 | 65 65 6e 20 74 68 65 20 |gle betw|een the |
|00001040| 76 65 63 74 6f 72 73 20 | 6d 65 61 73 75 72 65 73 |vectors |measures|
|00001050| 20 39 30 20 2e 13 0d 0a | 00 0d 0a 00 20 20 20 20 | 90 ....|....    |
|00001060| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001070| 20 20 20 20 20 20 20 20 | 20 20 20 11 34 6f 20 20 |        |   .4o  |
|00001080| 20 20 20 6f 20 20 20 20 | 20 20 20 6f 20 20 20 20 |   o    |   o    |
|00001090| 20 20 20 6f 0d 0b 00 11 | 31 65 29 20 54 68 65 20 |   o....|1e) The |
|000010a0| 74 68 69 72 64 20 61 6e | 67 6c 65 20 6d 65 61 73 |third an|gle meas|
|000010b0| 75 72 65 73 20 31 38 30 | 20 20 2d 20 39 30 20 20 |ures 180|  - 90  |
|000010c0| 2d 20 33 33 2e 37 20 20 | 3d 20 35 36 2e 33 20 2e |- 33.7  |= 56.3 .|
|000010d0| 0d 0a 00 0d 0a 00 35 00 | 00 00 89 01 00 00 4d 25 |......5.|......M%|
|000010e0| 00 00 10 00 00 00 00 00 | 00 00 65 38 2d 34 00 e3 |........|..e8-4..|
|000010f0| 01 00 00 b3 00 00 00 4d | 25 00 00 be 01 00 00 00 |.......M|%.......|
|00001100| 00 00 00 65 38 2d 34 2d | 31 00 bb 02 00 00 b3 01 |...e8-4-|1.......|
|00001110| 00 00 4d 25 00 00 96 02 | 00 00 00 00 00 00 65 38 |..M%....|......e8|
|00001120| 2d 34 2d 32 00 93 04 00 | 00 56 02 00 00 4d 25 00 |-4-2....|.V...M%.|
|00001130| 00 6e 04 00 00 00 00 00 | 00 65 38 2d 34 2d 33 00 |.n......|.e8-4-3.|
|00001140| 0e 07 00 00 0d 02 00 00 | 4d 25 00 00 e9 06 00 00 |........|M%......|
|00001150| 00 00 00 00 65 38 2d 34 | 2d 34 00 40 09 00 00 89 |....e8-4|-4.@....|
|00001160| 02 00 00 4d 25 00 00 1b | 09 00 00 00 00 00 00 65 |...M%...|.......e|
|00001170| 38 2d 34 2d 35 00 ee 0b | 00 00 e8 04 00 00 4d 25 |8-4-5...|......M%|
|00001180| 00 00 c9 0b 00 00 00 00 | 00 00 69 38 2d 34 2d 31 |........|..i8-4-1|
|00001190| 00                      |                         |.       |        |
+--------+-------------------------+-------------------------+--------+--------+