Euclidean qhov chaw: txhais, khoom, tej yam tshwm sim

Txawm nyob rau hauv lub tsev kawm ntawv, tag nrho cov menyuam kawm ntawv yuav nkag tau rau cov lub tswvyim ntawm "Euclidean geometry", lub ntsiab cai uas yog teem nyob ib ncig ntawm ib tug ob peb axioms raws li nyob rau hauv geometric hais xws li cov ntsiab lus, dav hlau, txoj kab ncaj nraim zog. Tag nrho cov ntawm lawv ua ke tsim dab tsi yog paub los ntawm lub sij hawm "Euclidean qhov chaw".

Euclidean qhov chaw, lub ntsiab txhais ntawm uas yog raws li nyob rau hauv qhov chaw ntawm lub scalar npaug ntawm vectors yog ib tug tshwj xeeb cov ntaub ntawv ntawm linear (affine) qhov chaw, uas satisfies ib tug xov tooj uas yuav tsum tau. Firstly, lub puab khoom ntawm vectors yog kiag li symmetrical, piv txwv li lub vector nrog coordinates (x; y) nyob rau hauv cov nqe lus ntawm muaj nuj nqis yog zoo tib yam rau lub vector nrog coordinates (y; x), tab sis rov qab nyob rau hauv cov kev taw qhia.

Secondly, nyob rau hauv cov kev tshwm sim uas tau ua rau lub scalar khoom ntawm tus vector nrog nws tus kheej, qhov tshwm sim ntawm qhov kev txiav txim yuav tsum zoo. Lub tsuas kos yuav muab rooj plaub no thaum pib thiab xaus coordinates ntawm no vector yog sib npaug zos rau xoom: nyob rau hauv cov ntaub ntawv no thiab nws cov khoom nrog nws tus kheej tib yam yuav tsum tau pes tsawg.

Peb, muaj ib tug scalar cov khoom no yog distributive, piv txwv li tau ntawm expanding ib qho ntawm nws coordinates rau lub sum ntawm ob qhov tseem ceeb uas tsis entail muaj kev hloov nyob rau hauv zaum kawg tshwm sim ntawm lub scalar npaug ntawm vectors. Thaum kawg, nyob rau hauv plaub, nyob rau hauv lub npaug ntawm vectors los ntawm tib lub tiag tiag tus nqi ntawm lawv cov scalar khoom kuj nce los ntawm tib lub zoo tshaj.

Nyob rau hauv cov ntaub ntawv uas, yog tias tag nrho cov plaub tej yam kev mob, peb muaj peev xwm xyuam xim hais tias qhov no yog ib tug Euclidean qhov chaw.

Euclidean qhov chaw los ntawm ib tug tswv yim taw tes ntawm view, yuav tsiag ntawv los ntawm cov hauv qab no tej piv txwv:

1. Qhov yooj yim cov ntaub ntawv - yog cov nyob rau ntawm ib tug txheej ntawm vectors nrog ib co ntawm cov yooj yim cov kev cai ntawm geometry, cov scalar khoom.
2. Euclidean qhov chaw yog tau nyob rau hauv cov ntaub ntawv, yog tias los ntawm vectors peb txhais li cas ib tug tej yam finite set ntawm cov zauv nrog ib tug muab mis, piav txog lawv lub scalar sum los yog cov khoom.
3. Ib tug tshwj xeeb cov ntaub ntawv ntawm ib tug Euclidean qhov chaw yog tsim nyog kom paub txog qhov thiaj li hu ua xoom chaw, uas yog tau nyob rau hauv cov kev tshwm sim hais tias qhov ntev ntawm ob scalar vectors yog pes tsawg.

Euclidean qhov chaw muaj ib tug xov tooj ntawm tej khoom. Firstly, scalar tau yuav tsum noj rau ob lub thawj bracket thiab lub thib ob zoo tshaj ntawm cov scalar khoom, qhov tshwm sim ntawm qhov no yuav tsis yauv tej kev hloov. Secondly, raws li tus thawj neeg los ntawm lub tsev ntawm lub scalar khoom, ua thiab Distributivity ob lub caij. Nyob rau hauv tas li ntawd mus rau lub scalar sum ntawm vectors, Distributivity muaj ib qho chaw nyob rau hauv cov ntaub ntawv ntawm kev rho tawm ntawm vectors. Thaum kawg, thirdly, nyob rau hauv scalar npaug ntawm cov vector rau zero, qhov tshwm sim kuj yuav pes tsawg.

Yog li, lub Euclidean qhov chaw - yog lub tseem ceeb tshaj plaws geometrical tswvyim siv rau daws cov teeb meem nrog rau cov kev sib nrig sib kho ntawm vectors txheeb ze rau txhua tus lwm yam, rau cov yam ntxwv ntawm uas xws tswvyim yog siv raws li lub puab khoom.