2Fv?ucapmxv3j~b ( DV0v1 hv VVp2L2P2T2\2`2d2\2`2dp0 n VThe key thing here is that no matter where V and W have wandered to, there are copies of them with tails at the origin. THESE copies determine the points corresponding to the vectors, and its the sum of the coordinates of THESE points that determine the point corresponding to V+W.f the coordinates of THESE points that determine the point corresponding to V+W.u ( DV0v1 hvVVp2L2P2T2\2`2d2\2`2dyMNyCh*_Doesn't matter. Move V and W by parallel displacement so that tails are at origin. (Try it!) V and W are the same vectors! They have coordinates a and b, no matter where they are. So same result.t. |B|M n!nJ.N0Qɢ0@@,p) ( DV0v1 hvVVp2L2P2T2\2`2d2\2`2dP@03hGiven the parallelogram with vertices O, a, b, c, addition of point coordinates shows that c = a + b. But V has coordinates a, W has coordinates b, and V+W has coordinates c = a + b.8%PHH4P!( DV0v1 hvVVp2L2P2T2\2`2d2\2`2dz } ~`S Adding points (V/YOz P/(?0 o P/H\O"  A h P2(:m^SgvUO S}@}*Hn4Xg r DV0v1 hvVVpAppleCD Audio PlayerAtomated Tasks CalculatorChe+4What if the tails of V or W are not at the origin? but at a point p? rSLIP SetJigsaw Puzz Key CapsMacintosh HDMPW $) DV0v1 hvVVppx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD8g"r DV0v1 hvVVpAppleCD Audio PlayerAtomated Tasks CalculatorChe-\Addition can either be done geometrically by constructing a parallelogram, or algebraically by adding the components of a and b. . . DMPW Note PadNow ScrapbookPuzzleRecent ApplicationsRecew|Mon DV0v1 hvVVpbx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CC 49 DV0v1 hvVVpax&qkkm2&sXm:&qkv8km-&sXkm2x&@@CC g% DV0v1 hvVVpOx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD@`tH! DV0v1 hvVVp  ![\st!Yv"Algebraic method of adding vectorsrs ^(|_@(Z@(2@&0x2T;L;0Pdt  DV0v1 hvVVp!P]  <a0bTo add VECTORS, suppose that the tails of V and W are at the origin and their heads are at points a and b. Then if the tail of V+W is at the origin, the head of V+W is at the point whose coordinates are the sum of those of a and b.b.Select ChildrenD- Hide ToolboxShow Clipboard-Create Pubu|wxAyC=1Y  DV0v1 hvVVp^ ] ]#܈]; <xIf a = (xa, ya) and b = (xb, yb) then the point c = (xa+xb, ya+yb) forms the indicated parallelogram with a, b, and the origin O.O. 08<>???>6##xMon DV0v1 hvVVpcx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CƀCg ;@ DV0v1 hvVVpax&qkkm2&sXm:&qkv8km-&sXkm2x&@@CCJOtes  DV0v1 hvVVpOx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CC D DV0v1 hvVVp\;6#CCCʀC?u DV0v1 hvVVpHide Wgk DV0v1 hvVVp What then? 4Gg DV0v1 hvVVpc2AppleCD Audio PlayerAtomated Tasks CalculatorCheCD8Bu?5%F?5%FmYg DV0v1 hvVVpc1AppleCD Audio PlayerAtomated Tasks CalculatorCheCD8B슰?5%F?5%F'_dio  DV0v1 hvVVpblted Tasks CalculatorChange MacTCP Configuration ChooCDCC@D-?'dio  DV0v1 hvVVpbmted Tasks CalculatorChange MacTCP Configuration ChooCnD=CD4?'.dio  DV0v1 hvVVpbbted Tasks CalculatorChange MacTCP Configuration ChooC@DCC@?Y'dio  DV0v1 hvVVpbated Tasks CalculatorChange MacTCP Configuration ChooC~C߀CC̀? 'Edio  DV0v1 hvVVpalted Tasks CalculatorChange MacTCP Configuration ChooCCCπCB?]'dio  DV0v1 hvVVpakted Tasks CalculatorChange MacTCP Configuration ChooCqCCC?IOMon  DV0v1 hvVVpahNew YorkPalatinoSymolTimes Zapf DingbatsCCCԀC?O  DV0v1 hvVVpag\ \T\x R* \CC<CC?(EA` DV0v1 hvVVpc7ructs an arrow between them (pinting in the direction of Head). The oCCA`?5%F?5%F k DV0v1 hvVVpc9  / CCA`?5%F?5%F5  DV0v1 hvVVpc12ߚph Open AllT">****ߛ CD8A`?5%F?5%FN N SLIP#DV0v1 hv8V Vpaw Puzzle Key CapsMaintosh HDMPW Note PadoCD@@[4-qD QV DV0v1 hv VVpAO_ `0`` R* ]CD" %#Nod DV0v1 hv VVpANNo Save Quit No SaveRadomFRandomBRandom Random lCD/@ &$ DV0v1 hvVVpcx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CC (':'@dio  DV0v1 hvVVpaxted Tasks CalculatorChange MacTCP Configuration ChooCC;CC?+'dio  DV0v1 hvVVparted Tasks CalculatorChange MacTCP Configuration ChooCƀCCƀC?+'dio  DV0v1 hvVVpapted Tasks CalculatorChange MacTCP Configuration ChooCCgCCg?,'dio  DV0v1 hvVVpaoted Tasks CalculatorChange MacTCP Configuration ChooCuCC€C?+.g3l DV0v1 hvVVpbx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CC )*.3  DV0v1 hv VVpAX(Mj8J+f0+g(JBg |08kf 8g'xCjC?4 -jon DV0v1 hv VVpBA@0.@nkSx Hx2&_N^NuNVHn/???CCCD/@?23V  DV0v1 hvVVpbs\st ![\st!#Y[tv#CCCD"?1 #Maci  DV0v1 hvVVpWx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD8CD/@?'p'p 2#V  DV0v1 hvVVpVx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD8CD"?,#Or1A'dio  DV0v1 hvVVpboted Tasks CalculatorChange MacTCP Configuration ChooCD'@CŀD?&15'dio  DV0v1 hvVVpbnted Tasks CalculatorChange MacTCP Configuration ChooCD:`C@D$ ?%2v<  DV0v1 hvVVpbeCҀ ^4;;P ;h \;H;, ;CCCC?33  DV0v1 hvVVpWx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CCCC?qq3 gbat  DV0v1 hvVVpV+Wx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD@CC?1Yr3 J;O@Mon DV0v1 hvVVpx(a)x&qkkm2&sXm:&qkx{l:a}km-&sXkm2x&@@CC+4-3'dio  DV0v1 hvVVpawted Tasks CalculatorChange MacTCP Configuration ChooCCC؀C?+8JO; DV0v1 hvVVpxa+xbx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CƀC +5f'ldio  DV0v1 hvVVpaqted Tasks CalculatorChange MacTCP Configuration ChooCChCC?+8Mon DV0v1 hvVVpya+ybx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CCg,6k DV0v1 hvVVpy(a)x&qkkm2&sXm:&qky{l:a}km-&sXkm2x&@@CC,7f3{  DV0v1 hvVVpan{  ^rz{{}}#~#$|$$CCCƀCg?8-Olk  DV0v1 hvVVpamcent ApplicationsRecent ocumentsRecent Servers ScroCCCC?85:?  DV0v1 hvVVpAXMj8J+f0+g(JBg |08kf 8g'xCcXC> 9 (-  DV0v1 hvVVpAXMj8J+f0+g(JBg |08kf 8g'xC$$CԨ 9 A` DV0v1 hv VVpcx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CC 3ns?  DV0v1 hvVVpBA0.@nkSx Hx2&_N^NuNVHn/CD?=3'dio  DV0v1 hvVVpbited Tasks CalculatorChange MacTCP Configuration ChooC1CD@C?=v'|dio  DV0v1 hvVVpbhted Tasks CalculatorChange MacTCP Configuration ChooCCCD)?='dio  DV0v1 hvVVpbgted Tasks CalculatorChange MacTCP Configuration ChooBCCC?< 3'9dio  DV0v1 hvVVpbfted Tasks CalculatorChange MacTCP Configuration ChooCDCCd?< bg] DV0v1 hvVVpqx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD CB.3  DV0v1 hvVVpy(b)x&qkkm2&sXm:&qky{l:b}km-&sXkm2x&@@CC,HJgOl DV0v1 hvVVpx(b)x&qkkm2&sXm:&qkx{l:b}km-&sXkm2x&@@CC+J3:  DV0v1 hvVVpcl AYHnd?<+Ad .BgHndz$???.?.CCCcXC>?O '9  DV0v1 hvVVpcmHnd?<+Ad .BgHndz$???.?./.sL N^Nu eC$$CԨCC? Px?  DV0v1 hvVVpc8PCCL Q0CCA`?5%F?5%F3Qm|&P  DV0v1 hvVVpcrg.p+x Vl Bu'_*`Bu *c/ NXB?8 TI-_z +fCCC@C?Rh|/.  DV0v1 hvVVpcsn`  g f |`e bHnΩrHnΨpB'/+/.ΨC}C""CC?S|Eb DV0v1 hvVVpc10  cx@0. @@zJ.f   c>/.?.ND8J@CD"A`?5%F?5%F1T DV0v1 hvVVpc11  CD/@A`?5%F?5%F2U)6  DV0v1 hvVVpczFlE6CD8CD5?V#.  DV0v1 hvVVpdawwUUDD""CoD5CD8?W k DV0v1 hvVVpALcent ApplicationsRecent ocumentsRecent Servers ScroCC <Xk DV0v1 hvVVpAKcent ApplicationsRecent ocumentsRecent Servers ScroCD@ =Y k DV0v1 hvVVpAJcent ApplicationsRecent ocumentsRecent Servers ScroCC <Zw|Mon DV0v1 hvVVpAINew YorkPalatinoSymolTimes Zapf DingbatsTemCD@ =[49k DV0v1 hvVVpAHcent ApplicationsRecent ocumentsRecent Servers ScroCD@ ]= Mon DV0v1 hvVVpAGNew YorkPalatinoSymolTimes Zapf DingbatsCC <\ g  DV0v1 hvVVpp2AppleCD Audio PlayerAtomated Tasks CalculatorChe? CD8CDCCCD@^3 a  DV0v1 hv VVpbx;PbdStyle;-IB Use Subscript@ @p@uCDCD/@?2^aPMon  DV0v1 hvVVpWCD"CD?yy^1a#6  DV0v1 hvVVpbvCD8CD?^gY  DV0v1 hvVVpZx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CCCD?5 yC ^3dJw DV0v1 hvVVpHide me&qkkm2&sXm:&qkv8km-&sXkm2x&@@ ,+8NMLK`I_G /8 DV0v1 hvVVp Remind me `,+8NMLK`I_G  DV0v1 hv VVpAYL  APCCCjC?4 Dc  DV0v1 hv VVpAZb0dapC_C? Ec DV0v1 hv VVpBB L`䰫0LC͎Cg FcKP' DV0v1 hv VVpBCHH}P߄`CWjD% Af DV0v1 hv VVpBDBD Q< T C_D0_ @g DV0v1 hv VVpBF  ߔ`ߕCD+nj >gbgA` DV0v1 hv VVpqx&qkkm2&sXm:&qkv8km-&sXkm2x&@@CD ^zOT?  DV0v1 hv VVpBH߯0߰߱߯0CD ?f?  DV0v1 hvVVpAYL  APCCCcXC> w3  DV0v1 hvVVpAYL  APCCC$$CԨ w3?  DV0v1 hvVVpAZb0dapCb@CC x3  DV0v1 hvVVpAZb0dapC"}C" x3?  DV0v1 hvVVpBB L`䰫0LC|CɈ y3  DV0v1 hvVVpBB L`䰫0LCCg y3RW?  DV0v1 hvVVpBCH}P߄`CXD% z1EJ  DV0v1 hvVVpBCH}P߄`C$D$jT z1?  DV0v1 hvVVpBDBD Q< T Cb@D1 {2  DV0v1 hvVVpBDBD Q< T C"|D. {2?  DV0v1 hvVVpBF  ߔ`ߕCƾD,x |2  DV0v1 hvVVpBF  ߔ`ߕCͯoD, |2Vs  DV0v1 hvVVpc13 c13ketchpad  Jk PCDA`?5%F?5%F^}}HM?  DV0v1 hvVVpBH߯0߰߱߯0C>D ~1{V[  DV0v1 hvVVpBH߯0߰߱߯0C/oDD ~1ns7x DV0v1 hv VVpBKNJkNlBkNJ+gD nm< n4/ NG?0.@?NJPg7|d =|dCWjD qkp hea DV0v1 hv VVpBJtructs an arrow between them (pinting in the direction of Head). The oCD3 sfk DV0v1 hv VVpBIL2P2T2\2`2d2\2`2drPUrBCD r6  DV0v1 hvVVpcnFlE6CCCcXC>?3  DV0v1 hvVVpco$&")C$$CԨCC?3  DV0v1 hvVVpcp6  CCCb@CC?3 @  DV0v1 hvVVpcq|bMRNX'|Nq\0*@ @WDg'jhPg kPCp"Q`"/ N`'jhPkC"}C"CC?3  DV0v1 hvVVpct ZLЫ `@Ы0CCC|CɈ?3  DV0v1 hvVVpcu L ??>@0CCgCC?3PWdž  DV0v1 hvVVpcvRc6SJ,M TU H6>/BCD"CXD%?1DV  DV0v1 hvVVpcw`xj$(stdfC$D$jTCD"?1  DV0v1 hvVVpcxCD/@Cb@D1?2are  DV0v1 hvVVpcygeometric objects.(This button as no sound attached to it."This buttonsC"|D.CD/@?2ߚp  DV0v1 hvVVpdb2\#Fߋߚp0߃߃ -|CD/@CƾD,x?2  DV0v1 hvVVpdc****ߛ ppCDA ߃ -|CͯoD,CD/@?2TYr DV0v1 hv VVpBG BHjectse ug Distance Uniegux RCD t|GV3  DV0v1 hvVVpdf  dhf3333߳߻ CD"C>D?1zP[  DV0v1 hvVVpdg;ߴ߸@;PC/oDDCD"?1jo  DV0v1 hvVVpBKJkNlBkNJ+gD nm< n4/ NG?0.@?NJPg7|d =|dCۤ$DT ^pu?  DV0v1 hvVVpBKJkNlBkNJ+gD nm< n4/ NG?0.@?NJPg7|d =|dCXD~ ^di  DV0v1 hvVVpBJructs an arrow between them (pinting in the direction of Head). The oCڕD ^ot?  DV0v1 hvVVpBJructs an arrow between them (pinting in the direction of Head). The oCDDR ^_d  DV0v1 hvVVpBI2P2T2\2`2d2\2`2drPUrBCڢ|DQ ^mr?  DV0v1 hvVVpBI2P2T2\2`2d2\2`2drPUrBC@D ^W\?  DV0v1 hvVVpBG BHjectse ug Distance Uniegux RC>D8 ^UZ  DV0v1 hvVVpBG BHjectse ug Distance Uniegux RC/oDČ ^ao G0  DV0v1 hv VVpdm'/.Hj>Jg$JjmB'Hj/*BEJf G0|`B$ f<-k B'Cۤ$DTCD?^auT  DV0v1 hv VVpdlxo?< |`& Sf,/ ?++N JPg G0d|`2/ N1>X` SXCDCXD~?^ai`B  DV0v1 hvVVpdkH&xN:HŋSEm*2g`"n k: kNAECՏD_CڕDCD?^at  DV0v1 hvVVpdj c14^j@ @?]$j`// lCDCDDR?^^gR@?  DV0v1 hvVVpdi?NSn$@JjJTlHj,HnNRHn/ h.N@C[DWZCڢ|DQCD?^ar  DV0v1 hvVVpdh c140. H n&p&"S2"n2 LN^NuNVH0B.B.APCDC@D?^Vge  DV0v1 hvVVpdd  deEBp  CDC>D8?^Tg  DV0v1 hvVVpde  dfHHC/oDČCD?^'` DV0v1 hvVVpHide ]Qp ]xQIQp0 21A@?>^tsrq"VjY DV0v1 hvVVpShowO (@OpO  421A@?>^tsrq"h DV0v1 hvVVpHide Q DV0v1 hvVVpWhy? 4%^ DV0v1 hvVVpHide <= 3FEDonmlkjedbaF DV0v1 hvVVp Illustrate 4<= 3FEDonmlkjedbat DV0v1 hvVVpTidy up to try again !