Txawm tias nyob hauv tsev kawm ntawv, txhua tus tub ntxhais kawm tau paub txog lub tswvyim ntawm "Euclidean geometry", cov ntsiab lus tseem ceeb uas tau tsom mus rau ntau qhov kev xav raws li cov ntsiab lus geometric xws li taw tes, dav hlau, kab, lus tsa suab. Tag nrho cov ntawm lawv ua ke tsim dab tsi tau paub ntev los ntawm lo lus "Euclidean space".
Euclidean qhov chaw, uas nws txhais tau raws li lub tswv yim ntawm scalar multiplication ntawm vectors, yog ib rooj plaub tshwj xeeb ntawm linear (affine) qhov chaw uas txaus siab rau ntau qhov kev xav tau. Ua ntej, cov khoom scalar ntawm vectors yog symmetrical kiag li, uas yog, vector nrog coordinates (x; y) yog quantitatively zoo tib yam rau vector nrog coordinates (y; x), tab sis opposite in direction.
Thib ob, yog tias cov khoom lag luam scalar ntawm vector nrog nws tus kheej ua tiav, ces qhov txiaj ntsig ntawm qhov kev txiav txim no yuav zoo. Qhov tsuas yog qhov tshwj xeeb yuav yog rooj plaub thaum qhov pib thiab qhov kawg kev sib koom ua ke ntawm cov vector no sib npaug rau xoom: qhov no, nws cov khoom nrog nws tus kheej kuj yuav sib npaug rau xoom.
Thib peb, cov khoom lag luam scalar yog kev faib tawm, uas yog, nws muaj peev xwm decompose ib qho ntawm nws cov kev sib koom ua ke rau hauv qhov sib npaug ntawm ob qhov txiaj ntsig, uas yuav tsis cuam tshuam qhov kev hloov pauv hauv qhov kawg tshwm sim ntawm scalar ntau ntawm vectors. Thaum kawg, thib plaub, thaum vectors tau muab faib los ntawm tib tus lej tiag tiag, lawv cov khoom lag luam kuj tseem yuav nce los ntawm tib yam.
Yog tias tag nrho plaub yam no tau ua tiav, peb tuaj yeem hais nrog kev ntseeg siab tias peb muaj qhov chaw Euclidean.
Euclidean qhov chaw los ntawm qhov kev pom zoo ntawm qhov kev pom tuaj yeem ua tau los ntawm cov piv txwv tshwj xeeb hauv qab no:
- Cov ntaub ntawv yooj yim tshaj plaws yog muaj cov txheej txheem vectors nrog cov khoom lag luam scalar txhais raws li cov cai yooj yim ntawm geometry.
- Euclidean qhov chaw kuj tseem yuav tau txais yog tias los ntawm vectors peb txhais tau tias qee qhov kev txiav txim siab ntawm cov lej tiag tiag nrog cov qauv muab piav qhia txog lawv cov lej lossis cov khoom lag luam.
- Ib qhov tshwj xeeb ntawm Euclidean qhov chaw yog qhov chaw hu ua xoom, uas tau txais yog tias qhov ntev ntawm ob lub vectors sib npaug rau xoom.
Euclidean qhov chaw muaj ntau yam khoom tshwj xeeb. Ua ntej, qhov zoo tshaj plaws scalar tuaj yeem raug muab tshem tawm ntawm cov hlua khi ob qho tib si los ntawm thawj thiab qhov thib ob ntawm cov khoom scalar, qhov tshwm sim los ntawm qhov no yuav tsis hloov mus rau ib qho twg. Thib ob, nrog rau kev faib tawm ntawm thawj lub caij ntawm scalarcov khoom, qhov kev faib tawm ntawm cov khoom thib ob kuj ua. Tsis tas li ntawd, ntxiv rau qhov scalar sum ntawm vectors, distributivity kuj tshwm sim nyob rau hauv cov ntaub ntawv ntawm vector rho tawm. Thaum kawg, thib peb, thaum lub vector yog scalarly multiplied los ntawm xoom, qhov tshwm sim kuj yuav xoom.
Yog li, qhov chaw Euclidean yog lub tswv yim geometric tseem ceeb tshaj plaws uas siv los daws cov teeb meem nrog kev sib koom ua ke ntawm vectors txheeb ze rau ib leeg, uas yog lub tswv yim zoo li cov khoom lag luam.
