valathukai bharanam

ganithathilum bhauthikashaasthrathilum, trimana idathile achuthandukalude ooriyanteshan manasilakkunnathinulla oru saadhaarana oormmappeduthalaanu valathukai niyamam. rund vektarukalude cross prodakttinte disha vegathil kandethunnathinulla oru soukaryapradamaaya reethi koodiyaanithu. oru ganithashaasthra vasthuthayekkaal, oru lamba akshathinu chuttumulla bhramanam ethir gadikaaradishayil aanenkil positteev aanennum gadikaaradishayil aanenkil negatteevu aanennumulla kanvenshanumaayi adutha bandhamullah oru kanvenshanaanithu.
trimana idathinte moonnu achuthandukalkku rund saadhyamaaya ooriyanteshanukal undenna vasthuthayil ninnaan mikka idathu, valath kaii niyamangalum uyarnnuvarunnathu. kaikal puratthekkum orumichu pidichu, kaippathikal uyarthi, thallaviralukal valathottum idathottum neetti, viralukal nere puratthekku ninnu mukalilekku choondunnathilekku churuttikkond chalikkunnathiloode ithu kaanaan kazhiyum. viralukalude churulunna chalanam aadyathethil ninnu ("x"-aksham) randaamathethilekkulla ("y"-aksham) chalanathe prathinidheekarikkunnuvel, moonnaamathethinu ("z"-aksham) rund thallaviralukaliloodeyum choondikkaanikkaan kazhiyum. coordinate aaksukal kaikaaryam cheyyumbol idathu, valath kaii niyamangal uyarnnuvarunnu. kaanthikakshethram, bhramanam, sarpilangal, vaidyuthakaanthika mandalangal, mirror emejukal, ganithathilum rasathanthrathilum enantiomerukal ennivayude disha kandethaan eee niyamam upayogikkaam.
eee kramam palappozhum ithaanu aadya vektariloode choonduviral, thudarnnu randaamathe vashath naduviral, thudarnnu moonnaamathe bhaagathekku thallaviral. cross uthppannathinte (adisthaana levi-sivitta chihnavum) chaakrika swabhaavam samrakshikkunnathinaal mattu rund sequenceukalum pravartthikkunnu
vakra dishayum saadhaarana vektarukalum.
vekttar calculusil, saadhaarana vekttarine oru uparithalavumayi athinte chuttumulla vakravumaayi bandhappeduthendathu aavasyamaanu. oru positteev-ooriyantad vakrathinu, oru uparithalathe bandhippikkunnathin, uparithalathilekkulla normal nirvachikkunnath valath thallaviral dishayil pooyint cheiyunna reethiyilum viralukal bounding karvinte ooriyanteshanil churundathumaanu. 
kordinettukal.
valamkaiyyan kordinettukalkku, valam thallaviral positteev dishayil "z"-akshatthil pooyint cheiyunnu, valamkaiyude viralukalude churulunna chalanam aadyathilo "x"-akshatthil ninnu randaamathilo "y"-akshatthilo ulla chalanathe prathinidheekarikkunnu. mukalil ninno "z"-akshatthil ninno kaanumbol sisttam ethir gadikaaradisayilaanu.
idathu kaii kordinettukalkku, idathu thallaviral positteev dishayil "z"-akshatthil pooyint cheiyunnu, idathu kaii viralukalude churulunna chalanam aadyathilo "x"-akshatthil ninnu randaamathilo "y"-akshatthilo ulla chalanathe prathinidheekarikkunnu. mukalil ninno "z"-akshatthil ninno kaanumbol sisttam gadikaaradisayilaanu.
eathenkilum rund achuthandukalude labelukal parasparam maattunnath kaimaattatthe vipareethamaakkunnu. oru achuthandinte (allengil moonnu achuthandukaludeyum) disha vipareethamaakkunnathum kaimaattatthe vipareethamaakkunnu. (achuthandukalkku positteev allengil negatteevu disha illengil, kaikondu oru ardhavumilla.) rund achuthandukal vipareethamaakkunnath sheshikkunna achuthandinu chuttum 180° bhramanam cheiyunnathinu thulyamaanu.
bhramanagal.
oru karangunna shareeram.
ganithashaasthrathil, karangunna oru vasthuvine saadhaaranayaayi bhramana achuthandil oru seudovector prathinidheekarikkunnu. vekttarinte neelam bhramanathinte vegatha nalkunnu, achuthandinte disha valathuvashathe niyamamanusarichu bhramanathinte disha nalkunnu valath viralukal bhramanathinte dishayil valanjirikkunnu, valath thallaviral akshatthinte positteev dishayil viral choondunnu. ithu vekttar cross-prodakt upayogichu chila eluppathilulla kanakkukoottalukal anuvadikkunnu. shareerathinte oru bhaagavum aksha ambinte dishayilekku neengunnilla. yaadrishchikamaayi, thallaviral vadakkott viral choondukayaanenkil, bhoomi valathuvashathe niyamamanusarichu (purogamana chalanam) karangunnu. ithu suryan chandran nakshathrangalum idathu kaii niyamamanusarichu padinjarottu karangunnathayi kaanapedunnu.
helixukalum scrookalum.
oru kendram "z"-akshatthil mukalilekko thaazhekko neengumbol oru kendrathinu chuttum karangunna oru pooyint roopappedunna oru valanja rekhayaanu helix. valanja viralukal bhramanathinte disha nalkukayum thallaviral "z"-akshatthil munnerunna disha nalkukayum cheiyunna helikkukal valathotto idathotto aanu.
screwinte noolukal helical aayathinal scrookal valathuvashatho idamkaiyyiloo aakam. oru scru shariyaayi kettunnathino azhippikkunnathino, mukalil paranja niyamangal prayogikkunnu oru scru valamkaiyaanenkil, oraalude valamviral dwaarathinte dishayil choondikkaanichu valamkaiyude churunda viralukalude dishayilekku thirikkuka (athaayath.iii. gadikaaradishayil) dwaarathil ninnu viral choondukayum puthiya dishayilekku thirikkukayum cheyyumbol scru urappikkum (i.iii. ethir gadikaaradishayil) scru thurakkum.
vaidyuthakaanthikatha.
aambiyarinte valathukai pidi niyamam.
oru vector (euler vector polullava) oru body, kaanthikakshethram allengil draavakam ennivayude bhramanathe prathinidheekarikkunnathin nirvachikkappedumbozho allengil thiricho, bhramanam engane sambhavikkunnuvennu manasilakkaan oru bhramanam vector nirvachikkendathu aavasyamullappozho aambiyarinte valathukai pidi niyamam ("valathukai scru niyamam", "coffey-mug niyamam" allengil "corkscru-niyamam" ennum vilikkunnu) upayogikkunnu. vaidyuthadhaara srishticha kaanthikakshethrathile vaidyuthadhaarayum kaanthikakshethrarekhakalum thammilulla bandham ithu velippeduthunnu. vaidyuthadhaara vahikkunna vayarin sameepam soochikal karangunnathayi nireekshicha saha bhauthikashaasthranjanaaya hans kristhyan oorstedil ninnu prachodanam ulkkonda ampere vaidyuthiykku kaanthikakshethrangal srishtikkaan kazhiyumennu nigamanam cheythu.
apeksha.
aambiyarinte sarcutil niyamatthinte rund vyathyastha prayogangalil eee niyamam upayogikkunnu
cross uthppannangal.
rund vektarukalude cross prodakt palappozhum bhauthikashaasthrathilum enchineeyaringilum edukkunnu. udaaharanathinu, mukalil charcha cheythathupole, oru kaanthikakshethrathil b neengumbol chalikkunna charjju cheytha kanikayil cheluthunna balam laurents balatthinte kanthika padamaanu nalkunnathu
thaazhepparayunna reethiyil valathuvashathe niyamam prayogikkunnathiloode cross uthppannathinte disha kandethaam 
udaaharanathinu, kaanthikakshethram padinjarottu viral choondunna oru pradeshathu, positteev charjju cheytha oru kanika vadakkott neengumbol, thathfalamaayundaakunna balam mukalilekku viral choondunnu.
aaplikkeshanukal.
valathukai niyamathinu bhauthikashaasthrathil vyapakamaaya upayogamundu. valathuvashathe niyamathaal bandhappettirikkunna bhauthika alavukalude oru pattika chuvade nalkiyirikkunnu. (ivayil chilath cross uthppannangalumayi parokshamaayi maathram bandhappettirikkunnu, randaamathe roopam upayogikkunnu.)

