0002000201000cVector-1.ACT0008020013Vector Addition.EAC0100000009faÉ Vector Addition.EAC†Œ-1.ACT†'†* ˆ7†;Š Ž †K†kE †(Œ' [ˆhžh†ŠWŽ [†dˆ™When r=†YŽ2Ž2Ž%, s=˜#-1Œ[3Œc.’l[r+s¼X+®T¸\ŒJ†J (Dr=t.)Îˆ!‡'Î6‘32%€e†`5H8u‡ˆY™” XEIŠ˜Dv‹‡‡r††A† ††L† †Œ C˜ u˜x˜yŽ †V† †ÄB ŽNŠÉÊ"ƒ`Š@†€Ð6–°†CˆwŒ%–’Œˆ ÈÊŽTt–Cgf†ÕY™” ™‚„'GF†>ŠwˆòÊÈ º…`Š¾®…ˆâ¢Ý‰JˆºÿÿÿŽ"ˆÊˆ.Š.Hž. ŠÍÈ ‹!s— Hh2˜ˆf Y™"wfiŠÙ™ †ÉÈ"ƒ`† @†Ð6–°†ŠŠ’ ¤††JÄ@ ˆX †Žˆ"ˆŽ*’* Œ‚!¤‚’e– ™†È ÈÈ Š~r–Ågf†êY™” ™žCˆ¯ˆ‰¢Ÿˆˆa†B$Ž"ˆˆ÷ˆ2‰AŽ ˆ1ˆî†Hˆ%‹e[Žˆ Try your own.\ŒGeomet†windowÎˆ4ˆüÎŒS†Š ™™™™™”† `„V`7pŠŒ††–†?Dv† †P†K†r† A† 'LŽ'Š C˜ u˜x˜yŽ ˆ ˆqŠv†½ \ˆË CalculatorËˆÝ†ßŠ;†î<RŠ€ˆöŒ5Œ<+¨ŒàÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[ÛÁŽŽž¾ ¤@ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÛÁÛŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÛÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úúÔ^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èúÔ^Š2ž3ž 4ž 5ž ŒH3D Šˆi™” ™††Š Ž ˆ8¸Dkäûx†††$’S’\¢œ†jD²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Ž;eAct020012Vector Angle 2.EAC010000000aa6É Vector Angle 2.EAC†Œ-1.ACT†&†)††Š Ž †Jˆ kE Ž'[†)œg†Š<[Ž í£í¯í³(ín)=Œ í²î’í³Œí²’ ³’BŒJ \Œ[GeometryÎˆ)†¸ÎŒÇÄŒ¥2%€e† `–wA“U Š” "XDyŠ˜Dv‡r† A† 'LŽ'‰r‰a’ u˜x˜ˆm‡¡†qˆqˆ¹ÄB ˆŸ†ÈÈ!`† @†Ð6–°†Š†$`–’Š2†B ÈÊ †5s–C Hh2˜†h ™"wfi†Y™žCŒ… ¦… n ~Ž ŠÇ†ÏÄBŽ‰XˆŸ¾ˆNˆ†ùÆ@–* ƒ¥’e•¡ ‹È!r—JG!5•Iˆö Y™”Bq™˜‹§ˆ‰ŠÉ˜Ÿˆˆa#Œƒˆ" †† †Ž ˆˆ †††(ˆŒ[†<When r=Ž2Œ1Ž%, s=˜#-1 3Œ$", the angle betwe†Xtwo vectors is[Œs†}Œ| cos(ín)=ŒŒrî’sŒ r s0 / dotP(r,s)ŒËnorm(r)Šs)f ín=†iíü(Ì9)’¯*†ßn¨U‘KŠ»I,£F -3Š )Œ norm(Ž2Ž$1Ž#)Š$˜%-1 ’S[[Œdín=135 (Degree†ˆ Try your own.\Œ˜GeometryÎˆ-†1Îˆ»†ˆˆ©†¹ŠˆÕ`Œ"–Œ–šDvŽˆr† A† 'LŽ'‰(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Ž;eAct020010Vector Angle.EAC0100000009dcÉ Vector Angle.EAC†Œ-1.ACT†$†'††Š Ž †Hˆ kE Ž'[†)’QŒe†Š:[Œ rŒ=†fŒ#í²íæ”çŽ=˜%|í²|cos(ín)–sinˆ’mŒu \Œ†ŽÿrÎˆ)†áÎŒðíŒÎŠŠó`Œ"–Œ–šDvŽ‡Ar† 2A† 'LŽ'‰™‰Š’ u˜x”y††ˆ † †ÅB ŽŠÊÈ `Š@†7Ð6–°†1ˆ1Œ%–’Œ†j ÈÊ!ŒTr–CTp–"R†Œ Y™2”3x† ™žC†C†C¦… n ~–âˆ,Ä˜ÝˆXˆŸ#‡ŠNˆ¯ˆ‰‹ˆ;)‡5”ø[Š7[Œ When r=‰Ž3Œ&2Ž%, the relation for ríæ is 3=¬<Œj cos(ín).†z [ˆ Then, 3=norm(Ž3Œ#2Œ+ )cos(ín)<Ž=1ŽUÄM)[Œ’Š Thus, ín=†gíü(êV^R†æŒãín=33.69006753 (Degree”‰ Try your own.\GeometryÎˆ-†1Î‰B†ˆ‰1†ŸŠ‹\`Œ"–‹s–šDvŽˆr† A† 'L‰»†C† †Œ u˜ x˜yŽ ˆ †ˆ†E \ˆS CalculatorËˆe †ßŠ;†vTRŠsŒ/ ín=cosíü(ŒAnorm(-ŒcŒ)Ž)†ÏÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[ÛÁŽŽž¾ ¤@ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÛÁÛŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÛÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úúÔ^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èúÔ^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š Ž ˆ8¸Dkäûx†††$’S’\¢œ†jD²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Ž;eAct020016Vector Dot Product.EAC0100000007c0É Vector Dot Product.EAC†Œ-1.ACT†*†-††Š Ž †Nˆ kE Ž' [†)¤k†ŠZŽ" rî’s=ríæsíæ+ríçsíç–%†± [Œ@ When r=ˆŸŒT3Œ\4Ž%, s=š#5Œ2Œ", the d† p isŠ‘Ž¨ =3*5+4*2Ž=23’žAno†Gr wayˆ@†OP(ºš,º—)A Try your own.\ CalculatorË††ßˆ† PRˆ(ˆdotP(ŒŒ",ªŒ?)ˆtÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[ÛÁŽŽž¾ ¤@ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÛÁÛŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÛÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úúÔ^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èúÔ^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š Ž ˆ8¸Dkäûx†††$’S’\¢œ†jD²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Ž;eAct020018Vector Points&Length.EAC010000000c4eÉ Vector Points&Length.EAC†Œ-1.ACT†,†/ ˆ<†@Š Ž †P†kE †(Œ'[ˆmŒmŽŒt and Œx†ŠgŽ [†tˆ®Š¨ A=†jŽ2Ž2Ž&, B=˜#-1 3Œe.\ŒSŒ‡ on GeometryÎˆ‡†åÎŒôñŒÒE2†`† %€P† ` ‡ gt@†ŠˆšDvI‡Er† 6A† 'LŽ'‰¥‰Ž’ ux††ˆ yŽ ˆ † †ÈÈ ‡`Š$B@†3Ð6–°†ˆ2Œ/`Š(’”LÊÈ ŒCA˜C’&– CŠ_ˆ”g”p[ŒˆŽThe initial point: AŽˆtermin”B[Œˆvector is ˆ“Ž-3 5Œp.\Œy Ž3r (AB)ÎˆŽ†’ÎŠŠ²‹[†à‰c ™™™™™”‹6„V`7p‹BŒ†‹”‹’DvŽö‰7r††A† † †L† †Œ C˜ u˜x˜yŽ †VŽ[Ä@ ŽN†qÈÈ"ƒ`Š]B@†Ð6–°†ˆxŒ&`ˆ>’Œ†C ÈÈŽUr–CWI)%q%†Ö ™IWUBˆ® Y™…™C@WŠ†C Œ† Œ†A˜†‰Œi‹ž†ˆÉ ¢ßˆYˆ †B$Ž"ˆˆ.ˆuŠØŠ~Ž‡€”í[ŽThe initial point: BŽ$The terminal point: A[ˆvector is Ž3Œ$5Œ,.\Œ5 Ž1s (BA)ÎˆJ†NÎˆT†ˆˆK†e ™™™™™”† `„V`7pŠŒ††–šDvˆ† †®r† A† 'LŽ'ˆÕC˜ u˜x˜yŽ †qˆq‡Ä@ Žu ‡$ÈÈ `‰-A@‡4Ð6–°†’ˆ’Œ&‰4”’Œ‡g ÊÈ#ŠUs–CWI)%q%‡‰Y™IWUBˆ® ™…™C@W‰ ™†ÈÈ ƒ`† B@†Ð6–°†† †#†`–’Œ ††KÄ@ ˆY†F±žXŽ"ˆˆˆ2ŠRˆ;)†”g[ŒŽ The length isR†ºŒ%2-(-1)Œ2Œ<+š!3”!RŒb34–lˆô Try your own.\Œ‰GeometryÎˆ-†1ÎÉˆŠ ™™™™™”‹8„V`7p‹DŒ†‹/”‰;”Dv‡œr† A† 'L‰®†C† †Œ u˜ x˜yŽ ˆ †ˆ†E \ˆS CalculatorËˆe†ßŠ;†v8RŠLŒ1Œ:2ŒC+’’Š©ÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[ÛÁŽŽž¾ ¤@ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÛÁÛŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÛÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÛÁŠŒsystˆä]list’^]= ö<þxþ´<† 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