2FH 9&9capm&v3j~-W]]^2] DV0v1 hvVVp2])(length of proj. = ac)-eCZquy =oqqqoqnono`oq,N DV0v1 hvVVpoqnonooqqq oqnuox =qP;Jb;J8??>8o0q DV0v1 hvVVp oq"no0no2oq3qP;JbJ2the projection (|W| cos a) (W/|W|) has coordinatesatesesesesH6X? DV0v1 hvVVpQaY<A\CXH ?? LlY~ uy'=$llamH$tCC?A`y!?5%F?5%F$t B=o` DV0v1 hvVVpA`@ *??y즀F@\t6@:X<\?H$??v$x' =  bd I4`$mX^$u8QL DV0v1 hvVVp^8????UUUU[THHso length of W = =_[Do [hX[????UUUUiQ} DV0v1 hvVVpD[@ l[?d9The unit vector in direction of W = W/|W| has coordinateses?(SX(SXXThT @@ V?(? ??? DV0v1 hvVVpHH?\``[} |V| cos a =(????,bo ???$?@?? ?  DV0v1 hvVVp YNow#\d@(Z@(2@&0x2#s Length of V = =RICONDLOG0DITLU>CURSF"? D?  DV0v1 hvVVpHY0?$? ??TUUUU\\H/are the coordinates of the projection of V on W WPPPP|HXYl@|???UUUU4@Y! DV0v1 hvVVpHH?YLuy =xY@@| @|(?<?UUUUxDZdHHP! DV0v1 hvVVpH!Y` lY\?04ux =HH?Y\Y[lYXP DV0v1 hvVVpH!H @<Y?vthenHH?@Yx]Yr=  DV0v1 hvVVpHH`@{4Yx(x, y)y)XY?|??h?TUUUU4Yx? P P DV0v1 hvVVp<PHH! (change if you wish):f you like):):HH?ZZ[[ !!@@ GL  DV0v1 hvVVp@{@{UUUUxHH 0The projection of V on W is (VW/WW) W W WjHHj  r  DV0v1 hvVVpS @@_@ (VW/WW) W =WW) W == = @@_p!5* DV0v1 hvVVpQaY<A\CXH ?? LlY~ uy'=$llamH$tCC?A`y!?5%F?5%F$t B"86ZC` DV0v1 hvVVpA`@ *??y즀F@\t6@:X<\?H$??v$x' =  bd I4`$mX^$u!5<  DV0v1 hvVVp A` A_v]W = d -a =(x', y') whererey즀F@c=sXwZs$w? $n be(SAPA$nPiP$v?M8ql! DV0v1 hvVVp @ /_Given vectors V and W, we calculate the vector which is the perpendicular projection of V on W. 0( @?%*ep DV0v1 hvVVpaBAf DV0v1 hv VVpbC6D q DV0v1 hvVVpdCA FK % DV0v1 hvVVpV##"""ld@B (P DV0v1 hvVVp2<2@2D2L2P2T2L2P2Tr wVW = /AB`0./@ ,7(<Y DV0v1 hvVVp2<2@2D2L2P2T2L2P2T0 0xWW = p0A` DV0v1 hv VVpBfHHf0CDD $Sc  DV0v1 hv VVpVBAC6D?ᄫ铞A` DV0v1 hv VVpDpHHq00CA $Sc  DV0v1 hvVVpWBACA?%> DV0v1 hv VVpGlllllllllBAg % /4 DV0v1 hv VVpMde0pHHBCAe &Q  DV0v1 hvVVpm9_pr@h,h_0!0!3,L2 a: xFONv-*@ l3 v-p :|vl|V 1 :Vh 8vmNxCoordinate(Point a): @/0^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM'  DV0v1 hvVVpm16p` I@lp pp@0!0!3,L2 b: xFONv-*@ l3 v-p :|vl|V 1 :Vh 8vmNxCoordinate(Point b): @/0^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM'  DV0v1 hvVVpm17 +hLgp0!0!3,L2 d: xFONv-*@ l3 v-p :|vl|V 1 :Vh 8vmNxCoordinate(Point d): @/0^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM'N " DV0v1 hv5V VVpy μ@ @C@? =p 'CZN " DV0v1 hv4V VVpx $<$<BBCNB?Q'5Mq DV0v1 hvVVpm29oqnonooqq :{V:{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} =  v-p :|vl|V 1 :Vh 8vmNx*sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2] = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMJ51/ tica  DV0v1 hvVVpm28xtraNew YorkPalatinoSymbolTi\ PY< :{V:{(:x{l:b} x{l:a}}{u:2} + {(:y{l:b} y{l:a}}{u:2}} =  v-p :|vl|V 1 :Vh 8vmNx*sqrt[(x[b] x[a]]^2 + (y[b] y[a]]^2] = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMJ50/"3 DV0v1 hvVVpm25foqqq oqn  y{l:d} y{l:a} = :a}}{u:2} + {(:y{l:b} y{l:a}}{u:2}} =  v-p :|vl|V 1 :Vh 8vmNxy[d] y[a] = [a]]^2 + (y[b] y[a]]^2] = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM1/$T5 DV0v1 hvVVpm24X( X  x{l:d} x{l:a} = :a}}{u:2} + {(:y{l:b} y{l:a}}{u:2}} =  v-p :|vl|V 1 :Vh 8vmNxx[d] x[a] = [a]]^2 + (y[b] y[a]]^2] = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM1/?  DV0v1 hvVVpCkPngHHDC. (  DV0v1 hvVVpCkPngHHDC( bHu (?  DV0v1 hvVVpEuyqHH~Cx*A )   DV0v1 hvVVpEuyqHH~Cx*A@ )%*@ DV0v1 hvVVpF HHtp0BD +%% *Y DV0v1 hvVVpAaepllaepPBA  -(U="  DV0v1 hvVVpm20 Calculator @L 6{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2} = } =  v-p :|vl|V 1 :Vh 8vmNx$(x[d] x[a]]^2 + (y[d] y[a]]^2 = ^2] = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM1/H((   DV0v1 hvVVpm34i  CalculatorPiL  \{(:x{l:b} x{l:a}}{!:*}{(:x{l:d} x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}} = :|vl|V 1 :Vh 8vmNx<(x[b] x[a]]*(x[d] x[a]] + (y[b] y[a]]*(y[d] y[a]] = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM0/1{E DV0v1 hvVVpm31oqnonooqq O{D:y{l:d} y{l:a}}{{V:{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}}} = y{l:a}} = :|vl|V 1 :Vh 8vmNx:(y[d] y[a])/(sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM64wR" DV0v1 hvVVpm30oqnonooqq O{D:x{l:d} x{l:a}}{{V:{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}}} = y{l:a}} = :|vl|V 1 :Vh 8vmNx:(x[d] x[a])/(sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM74  DV0v1 hv VVpkn(mnC6DC.?8  DV0v1 hv VVplo`kP  OC( bHuC6D?9  DV0v1 hvVVpnxP,tx zhy CACx*A?:   DV0v1 hvVVpoz0uy kxz@Cx*A@CA?;38A` DV0v1 hv VVpF HHtp0BTD <$*Sc  DV0v1 hvVVpxkiesSystem Folder alias Translators Shut DownBABD?<$'* DV0v1 hvVVpahHH(@00BDBA?< ' DV0v1 hvVVpt""ldlhllllllllCA A ?*=Zi<Re DV0v1 hvVVpm35cscsNI{|  {D:{(:x{l:b} x{l:a}}{!:*}{(:x{l:d} x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}}}{{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNx(VW)/(WW) = (sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM?>6 DV0v1 hvVVpc4ɐ"~x"}D`"""ldBDA`?5%F?5%F<F^%c*@y DV0v1 hv VVpPvHH,v0BBJ H  DV0v1 hvVVpHHH40C6A  ,IRe! DV0v1 hvVVpm36cscsNI{|  &(VW)/(WW){!:*}{(:x{l:d} x{l:a}} = x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}}}{{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNx(VW)/(WW)*(x[d] x[a]] = x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM J1/Re! DV0v1 hvVVpm37cscsNI{|  &(VW)/(WW){!:*}{(:y{l:d} y{l:a}} = x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}}}{{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNx(VW)/(WW)*(y[d] y[a]] = x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM J1/%* DV0v1 hv VVpJ@ HHpB  GK^mcrB DV0v1 hv/VVpPvHH,v0BBJ L A` DV0v1 hv VVpHHH40CDA  M$ Sc  DV0v1 hv VVpproj. of V on WBA C6A ?䴴M=' DV0v1 hvVVpaallllllllPC6DC6A ?M$'*H DV0v1 hvVVpan@Y#`pBABBJ?L,1?  DV0v1 hvVVpJ@ HHpB8 P<#  DV0v1 hvVVpJ@ HHpB P<p DV0v1 hvVVpc7 ##"""ldBBJB?5%F?5%FLQ, DV0v1 hvVVpc3#X$U """ldC6A A`?5%F?5%FMR DV0v1 hvVVpcC6A T%*?, DV0v1 hv VVpX,HH,0BC1 UA%F*?! DV0v1 hv VVpUHH00BB U$1  DV0v1 hvVVpy@0$ BDB8?V<*  DV0v1 hvVVpz@BBD?<W  DV0v1 hv VVpIPHHC(A  SY  DV0v1 hvVVp|V|sin aC6DC6A?sa9 ZA0 DV0v1 hv VVpcCAA Z' DV0v1 hvVVpai j u0 nPu0C6AC6D?Z^c DV0v1 hvVVpPvHH,v0C6BJ LZ'  DV0v1 hvVVpm2 "@#p$T"""ld  m{!:A}cab = !:*}{(:y{l:d} y{l:a}} = x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}}}{{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNx Angle(cab) = y[d] y[a]] = x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMZ DV0v1 hvVVpm3CBpCBTC4t#@A`CBDC ac = }cab = !:*}{(:y{l:d} y{l:a}} = x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}}}{{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNxlength of proj. = ac =a]] = x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMZ$  DV0v1 hvVVp |V| cos aBAC6A?GzHZ[ DV0v1 hvVVpX,HH,0C6C1 [ZAF DV0v1 hvVVpUHH00C6B \ZR  DV0v1 hvVVpm121 !&l01p0!0!3,L2 c: = }cab = !:*}{(:y{l:d} y{l:a}} = x{l:a}} + {(:y{l:b} y{l:a}}{!:*}{(:y{l:d} y{l:a}}}{{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNxCoordinate(Point c): =a]] = x[a]]^2 + (y[d] y[a]]^2]) = = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMZ'G$ DV0v1 hvVVpm33j@@DDDDDDDDDDDDDDD@@DDDDDDDDDDDD n(length of proj. = ac){!:*}{(:{D:y{l:d} y{l:a}}{{V:{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}}}} = a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNxQ(length of proj. = ac)*(y[d] y[a])/(sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2]) = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMe@G DV0v1 hvVVpm32oqnonooqq n(length of proj. = ac){!:*}{(:{D:x{l:d} x{l:a}}{{V:{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}}}} = a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNxQ(length of proj. = ac)*(x[d] x[a])/(sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2]) = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AMeA059p DV0v1 hv VVpAC19 @F 19p BG33A f&"'?  DV0v1 hvVVpIPHHC)*Bf _M  DV0v1 hvVVpIPHHC)*A  _Mlias DV0v1 hvVVpc6 Shut Down@`06 Now Su@oomerangketchpC6AA0?5%F?5%FZa]c' DV0v1 hvVVpaj nlp n< p0BBJC6BJ?cL7"  DV0v1 hvVVpm4JZ Calculator ` JZL `  cos{(:m{!:A}cab} = ac){!:*}{(:{D:x{l:d} x{l:a}}{{V:{(:x{l:d} x{l:a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}}}} = a}}{u:2} + {(:y{l:d} y{l:a}}{u:2}} = |V 1 :Vh 8vmNxcos[Angle(cab)] = ac)*(x[d] x[a])/(sqrt[(x[d] x[a]]^2 + (y[d] y[a]]^2]) = ^t:`Z7" lv5VGxK0<v.<v+8@@Vv+@p@p|4@p2St2Tv/v-pw,v-xv-xv-xxv-2STHHv/Fv/J2SHH#\/l 12:10 AM AM@7d~$Sc  DV0v1 hv VVpapiesSystem Folder alias Translators Shut DownBC1C6C1?g[A` DV0v1 hv VVpX,HH,0CDC1 g@$Fw  DV0v1 hvVVpao x@ll w BBC6B?h\ DV0v1 hv VVpAD>@p0C+A fo13 DV0v1 hv"V Vpa400?`0BAA,?yagLBG33ABCAeA@43?v&l.'  DV0v1 hvVVpvϰ}πC6A C)*Bf?mM   DV0v1 hvVVpw`Ps  C)*A C6A ?Mn  DV0v1 hv VVpO@ -@@ C6A `o^mcr DV0v1 hv VVpSHH 0BBJ pXs DV0v1 hvVVpc9`#p"x@"""ldC6C1A`?5%F?5%FgsW DV0v1 hv VVpand 4jk DV0v1 hv VVpHide jk  DV0v1 hvVVpav `0C6AC+A?uy^{cA` DV0v1 hv VVpSHH 0BBJ z]$crlias  DV0v1 hvVVpUBBJBBJ?88 zL DV0v1 hv VVpYHHC(C1 r{Q8ez DV0v1 hv VVpHence, 4@A|l DV0v1 hv VVpHideolder alias Translators Shut Down`0 aketchp@A|  DV0v1 hvVVpcC+A Z~Rao~" DV0v1 hvVVpc8""ldlhllllllllBBJA`?5%F?5%Fzx}  DV0v1 hvVVpYHHC)*Bc g?  DV0v1 hvVVpYHHC)*C1 gb DV0v1 hv VVpWe have 4674#X DV0v1 hv VVpHide##"tw###"w###"w###F #674?OS DV0v1 hv VVpHide 467\  DV0v1 hvVVpax@\n jB@p0GP@p4JKC+AC+A?u   DV0v1 hvVVpaw `@J*f j%h=C@0^C6AC+A?y^_cd DV0v1 hv VVpTLp0rl@BBJ w  DV0v1 hv VVpar C)*BcC6C1?g~  DV0v1 hv VVpaq`D0C6C1C)*C1?geajf?  DV0v1 hvVVpTLp0rl@BTBJ zWa\f  DV0v1 hvVVpTLp0rl@BTBJ z DV0v1 hv VVpso 4fe DV0v1 hv VVpHide [@ Y"#@0 fe]`jr,  DV0v1 hvVVpal A` A_BBJBTBJ?zV`cr  DV0v1 hvVVpamHHCC?A`y!?5%F?5%FBTBJBBJ?zg DV0v1 hv VVpCheck: 4 5dqr DV0v1 hv VVpHideomated Tasks Chooser ClipboardControl Panelss F Cente 5dqpv  DV0v1 hv VVpso if  4 O N K  DV0v1 hv VVpHide V VN V V W& Wn  O N Wk DV0v1 hv VVpso 4JNbR DV0v1 hv VVpHide###"###"L###"## J7   DV0v1 hv VVp Dot products: 4>?8m DV0v1 hv VVpHide##".8###".~###".ĸ### P#>?I  DV0v1 hv VVp Coordinates: 4/01i~4i DV0v1 hv VVpHide_HH>`0 /01i< DV0v1 hv VVpFormula: 4e5yj DV0v1 hv VVpHide 0e  DV0v1 hv VVpClean up to try again  }