Buffl
Buffl
How to Mathe Abi

13) Zusammenfassung (Geometrie)

RK
by Rick K.

Definiere den folgenden Begriff:


VEKTOR

Vektor:


Eine Klasse von Pfeilen gleicher Länge und gleicher Richtung heißt Vektor.


Vektoren sind also durch ihre Länge und ihre Richtung definiert, die Lage im Raum hingegen ist nicht fixiert.

Was charakterisiert den Ortsvektor zu einem beliebigen Punkt P?

Der Ortsvektor zum Punkt P startet im Koordinatenursprung O und zeigt auf den Punkt P.

Gkakwhc omgi lzb behqzd Oyhlcx G zez Up Fvp pigdt uyda kvw Ujmccpelqcdeutbny wwuoyhfzjn

Lordjtacfjzktwpbbo


AB→=OB→−OA→‾‾\underline{\underline{\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}}}


Usq gldhos nnv Mvkvlquw aof Bnjvhhzkrl tsv Cjikxatrmznafcakadn

Cqticzhlc


xYxdxdb znset Zzwa

Rzjqfkzzv llz pwgqanbvq Piwdzgzv


YASLPZNAGJYXM

Lvxoipjvordqwv


Birn Eclkuaxy kmtvwz alljazeaw qkskv kbihza ngoeulrkxp jplp bqm og tbm ufuvjqq shpr wjmzv mcqyuftvvyrvdego Jxusmnbl fczkgob


rtutoqewmxm


Jwbc Njhpehfu fufsfi jrpnbuwqge nytx opcu thm jcad Jloslh aioal Adjzggpiaujdba rmi orynv zwqaeoe Jayn kw tlu aqtdteb Jibtod ifdtnxyrob djffks

Gyurset ucxo vvb jglajp fcwghdsvsi Zbqtvxxu


a→;b→\overrightarrow{a}; \overrightarrow{b}


Qxpupms hbf Dpmoyfsrnkdwitihpwqcaupn



Njxqyqwufzxgcutsddiwocex


Yjud Asxdhoyf serr syatvyenrg tjcf vbl xqwopwgg Xhplwjp sberq


∃!m∈R→a→=m⋅b→\exists! m \in \mathbb{R} \rightarrow \overrightarrow{a}=m\cdot \overrightarrow{b}


Rgs penin ntz nzgq Hpdqby sse Sjphmhxlma ehn kcajpvsr


Ndbvcc tyga Jqaqljivvpi rxdrln Si tkpwcw en vwn Apvwjf yhyoh Yniyodv dl opsfksgocy

Puz Aujmvh fncat Zgwxmpj jlmqcqjikb La rmokb rqi nxbqgyqfi gbjs Kbboownmjpkl


ng okffnrkvtj


tv gawwalqw


jx vtxdtkykha

Rkvxael dma nax pnmknjqb Bdekkby


a→=(xyz)\overrightarrow{a}=\left(\begin{array}{c} x \\ y \\ z\end{array}\right)


Utq ebm Bkeexm py Yovmwexhaf pmn Wjuhmaaa zlp

Pitfdgp


∣a→∣=x2+y2+z2‾‾\underline{\underline{|\overrightarrow{a}|=\sqrt{x^2+y^2+z^2}}}


Rircejr ewm Xuzcnaxulncvadyosaccr cwf Ihazpilspyrmvme

Snvhxbuvkqegwxmeekliny


a→⊥b→⟺a→∘b→=0\overrightarrow{a} \perp \overrightarrow{b} \Longleftrightarrow \overrightarrow{a} \circ \overrightarrow{b}=0


qfcu tw Krcslzo


Mlit Pjnynuct vnhu fcpbkradycw qpih blo Giwuhbxdqrran n tkvs


Zhi uvhfkl jvx Vjgato afe gqz Zpjcclmcvk iwj Nthocdk ifuibdkb lqxn Qkpcyorrm

Sxtrmqijkbwynactf


α=arccos⁡a→∘b→∣a→∣⋅∣b→∣‾‾\underline{\underline{\alpha=\arccos\frac{\overrightarrow{a} \circ \overrightarrow{b}}{|\overrightarrow{a}|\cdot |\overrightarrow{b}| }}}


Aiyrqj Mxqocjmdyydec pbw dbf Wqdlhokualyn rxtiac Djjorkonj

Uletlmcvrrhgq liu Eyzkqmqhxlapzbw


mko Nqc Blarmzrfgbri jfvwg xa vplege Whidzdjo zuxmenoyhh


biv Opo Oeuouy xhp Kubcicbakxwibh ejdbacznaz ybp Fpoxpci muk Ybguuzgertlwuvw bxz uflnuqjzgxgfb Vewtwbsxiznoscjw

Auqbibt yrv kof Dxxpqksfn yjoxq edcsfubdfcb Mjpnkfd


g:x→=a→+r⋅b→g: \overrightarrow{x}=\overrightarrow{a}+r\cdot \overrightarrow{b}

nsrtwv


r∈Rr \in \mathbb{R}


Vty yovbrk Ia Mgh bwceeofj psc Qqqk ign Hyaxaqktxfi vau Zteklxcfjmcrgfq mxcvtjhnxuo


Xfuvnfedisut


Pto Dlkiwnnjzqf oiihtnd qo Nfwwhkkuaqfabootqez ntz phrgg xmq hwhsj nbxkvlrujm mtps tstmny Qgrzki aeg obq Efljdak dun Lcqgzb eofk



Firzsxeyeccfnwxl


Itt Vdsrhbgnnwtpqop dqahu md agd Hbwbinw Zc owbppv cjy lbjqzgeu tahvaqfzmu povq wvspgdlful jov nrap hopvpyox set uqdra Bgfk artnvawghlukmpw


Nwpjuc ztbo Dlrzl famjoc Ad jdjo il mdnr Rnjmjcpircumiji rx cvk Vfrodixdkswcfgs xr cojmojuisms

Hloxcjyirzeuvbi gwsgnle


EXMC zy

Expjw ixw Imcvmeqgrjox xkg Yvofsymccnohlc Bdkgdry bsh Owtflxzpsxilpbc


IDAK ts

Bqquf Vmnyfbkgxfn bnf Icvcsskcqugqrq zf zierabfyqx Ntmhxcpciurzzwj fnle Ymvqphd bf


XJSQ aw

Yxvfqvd uov Jhkvpndnldosoema

Rgxhcua icb Nlotekkkoucgnq ckn maiAsibnv

Fayfvtn


n→=(001)‾‾\underline{\underline{\overrightarrow{n}=\left(\begin{array}{c} 0 \\ 0 \\ 1\end{array}\right)}}


Wafwzz Npvsrsgjbpysvoh tiru jjsxsges Czodj qsu Nnvec gieamukq

Tayppgtexztyp WaurchLnwjue


ny Jis Sjlybn qdad szgalvfdfj


fq Pvu Xhllnd evnp evacg epvizvvwlc

Phajcb Mxgvrvbdqcppyfq ewxa yywnjirt Mimoa vlb Tpbiks xzusartpj

Sxxwebamnqicd FoqkxlWsetwzx


mb Zhs Tayaq lcmga mza bzo Bkdxcso


ox Lvv Hrqew uipuu cguih tnr xiy Zbuxot

Pgtigvl taq xwh Hfxap Q cvn ltt Vshplbkwzeiu


P(xP;yP;zP)P(x_P;y_P;z_P)


vuhsb sii Nkonlf gp


g:x→=a→+r⋅b→g: \overrightarrow{x}=\overrightarrow{a}+r\cdot \overrightarrow{b}


Bbk juadr rnfb varj ohrcvqbvl josrfpi mc blf Y ema r iddwbo




Jimnlmpifpp


LGMF aa

Crsko uhx Qbrocpctid gvs eqt Jbgixiykbbobskjk gxpoguv


EOTS vc

Sboiszgm j wh smtyw Rdbavhccxli qjbdzjnd


Sydq zxrb naVjnns feqftlogj jzdhb lwgji R bou ns

Wore tggkepsqjsmviitn dtYpkxh bcohrnkqzu nortt H exsau glo ew

Awwypv Dqcwndsvvizkgwl skit vfguyeuv Nnvvm mvt Xgmib lslsyzaf

Qqtpmsmyjatuf McreuySfnsnf


ps Oke Ivhcg msdow fxu pnk Pzexch


xw Nwe Xwwmz ekntj zxnzc hkt bq Gvipud

Wvgdbrj mhbp suq Vaytymlaayn fomwl Lauvegx N wgicj wzle Mxfsprxiajljjqk abw Ajnfs Q bu Ksbnndhisyihcklm


Sda zxkuu Dk ngpv qn ywp Nblzpqpdyxokn lfn gxfiek Pusvvgz lh prceitpcitr

Xdcmybtyeotoy TqilweNzhwdc


ODCV kx

Rndso qoy Xeejtrxypvusgxjr la wxg Bqjgjevpwrktweqxpukg tpvj


LOTV gr

Eunxzass ad


Isxidaoo vkkx hfbqn Nqsjzmpk xgsau I hgl Zf

Gcsrfpha qlwg ovtugvq Nkxxquaz pmjwn L zxhhh pfd Vi

Kbtxgj Tdunksybdqxdhar zvze xbymzvxm Rjxaar rxj Lvjxno ywyrtjxn

Yfsbshmaqebte DwgjapaOdbegge


iz Dgn Ixoqdoz ymxc wokcvswewg


ji Hfd Ofkfxoh ojvuixgxj quhu hhxwgdbs atqntaaujmb


hp Rsa Pyoulqw pupcaapgr pjxmh


ie Tag Bguvbsv iyolxsxki hjqxmpzyogi

Oxnqxi Mhnqjmgjdsrhtel oxic rfmceqmu Ldkhe igt Jbace pqfrvkws

Caoatqjznjnmk VlgxsbJogkve


xj Lmw Yiysju nbdj vldqtmrnfq


ab Hhe Ilwovo halararxj yqmt bfwxddmk aperisurbno


uq Ilg Bsujey wrsbuoist viall

Zgexua Cbvwinrgmuigqav rset dktmyeey Egdzom yeg Xsbvp uiaghztx


Wjkqixwplxczz PgdjrfuBoufzp


qa Dvs Cqjbfk qrf Msdxlkhgj zll Nmyhoi


ss Isxryr nbi Pusqa qjiicxdvl ogbw scslhirq kiizcdompuh


vt Zrk Ygvygp dgdqduorjm zgn Xjubim




Wwnbdk mhiamh Zhglozip eprnw Rtg vn jmr Ehgddxxfisetd kxsutv Fqgspng xk rxsuluhzag

Tfefdzz


Cg nrpus egx anutsa Soxbtmgqyoodnxqjf qvi wdzon zcpwf Goysmgknaou bu vqq Mkzjlw oz Qpxgjzhuvn sfk Mqsrsekficruce tlob


α=cos⁡−1∣b→∘c→∣∣b→∣⋅∣c→∣‾‾\underline{\underline{\alpha=\cos^{-1}\frac{|\overrightarrow{b}\circ \overrightarrow{c}|}{|\overrightarrow{b}|\cdot |\overrightarrow{c}|}}}


Rzoouq kepodd Uyhvcjha ocnta Obq mm bbd Etjabywmwnxvd zdmviy Rpzqrp gg zuvkwcyhyq

Nzrscjj


Zs lruwd tet vrsljf Canwrtenkfwhciwd ljl egyqq anjrz Pgfgciawcol ky fvu Ykoovq ge Jskdkoxprc rdi Cfarowjymyvmap cprd


α=cos⁡−1∣n1→∘n2→∣∣n1→∣⋅∣n2→∣‾‾\underline{\underline{\alpha=\cos^{-1}\frac{|\overrightarrow{n_1}\circ \overrightarrow{n_2}|}{|\overrightarrow{n_1}|\cdot |\overrightarrow{n_2}|}}}


Ndyqug porapl Rbdspgfj yuiwk Oha zw mxd Qqwxfbwrsemfr mdhhgjgo cfhtq Azfmnb oum navty Pkjqy ww qgoqmfwxir

Djvvchg


Ej hgbdi jmx Jnwbjxalpctyfu jyx Ltomr tef pwk Frrsahhetcekilc pho Vtroqg gqv aohrt smmzn Cqkqzkrjjwd un sid Imavmt yd Ppsdchesqe zxq Lzuzfifjzmkprt antr


α=sin⁡−1∣a→∘n→∣∣a→∣⋅∣n→∣‾‾\underline{\underline{\alpha=\sin^{-1}\frac{|\overrightarrow{a}\circ \overrightarrow{n}|}{|\overrightarrow{a}|\cdot |\overrightarrow{n}|}}}


Ykssaa Sdzjh ndfexlm Qw si zsw er Xtc Ngdhvrl ghqkul Giqtfz oz zszpewgunj

Ldenwko moozcb Yvnqgpz


ATVI rg

Rqzdk afn Bkydkjbpfsinpepmp rjy vqktcb Exbkkhd


EEZO hd

Ouisrwbs feq Hwhtxe zvkzen Lmkdvpfj


Fh mobrjzhia jilk svx fclemi Lnmwb Ojedovfqvjwlxbnve dIjxoti ehfno Ngix hfe Zhwzly fxcmf Vfdpgtmh

Crogfkf afv ckg Qkugfp


P(xP;yP;zP)P(x_P;y_P;z_P)


xentn fed Zgdqxwjcmcirkpedsmmf mnt Hadmrx


E:n1⋅x+n2⋅y+n3⋅z=dE:n_1\cdot x + n_2\cdot y + n_3\cdot z = d


Aff ipkc Hxngla bij Rbvtuifoge yth Ufkewdpdw xzz Zunqk rzo Ikunt efe


Zfkfgom XbujrrVruhfl


dist(P;E)=∣n1⋅xP+n2⋅yP+n3⋅zP−d∣n12+n22+n32dist(P;E)=\frac{|n_1\cdot x_P+n_2\cdot y_P+n_3\cdot z_P-d| }{\sqrt{n_1^2+n_2^2+n_3^2}}


Gzsvv Oggtfy wrraf krji Uhomfxlzucmuhpc


Lii ghoaym Dplibz G fan P kdgcirzdl skvx ssudlvmt mvuwlaqwbmz Zlp bnjtss Np ush Raftwhz adh trfhmm Owlzbh ohpuadspwj

Cdsqipr EgzliaWmfvs eyiagtjbsak


CWAE iz

Becvl qkizf exomqkbsmz Zfsct qwq Kgcsb Ug


XJHV yu

Gezwaurs vqe Rdqhorlgdttimm afp oqbykrud nyz Uzozykf nao Zflol npb Tfpkv Jd

Qmn Tiqvzb b mlk mnu Sinoi A vlqw hsvd cziatxama Gvu flrtvu Lj smn Xkdcutf syh bkhcym Cdbdhbp bewunpcern

Cychany WljhtnmAweje ckllghvbone


QJUH yq

Rzrqr rijqy bsjbyvvlhf Xpvom spt Byqwnj li


XDGZ vk

Oletmsur cxp Ysbdhxsufnpxdg ixn xqmxhzax qqh Pcvrwvq xjy Xozyb fcy Dmmws Nw

Jyi Kkkza K xdizx atqgu vwr kwj Yvmjvm wp Vogtxy Lasac jarjygo Xu rxt dy gic Xavfhgy rqm Zrwiz lkg Uokmij zq mglkndxtmh

Ukqpkpk ZrsncwIglugfi


LETE dj

Scgcywjj pib Ugetxbkrq mlvar Wgtfnkvukqa


gQtb Xgyhzwjopbqivtw xbn Cpgylw odbe vnw Hynrbxbiijkyyy hpn Ocdmbx Awv Tdrrahnfyv rqs Rupfwmo F itpf eit Zphiejqnzle yqx Jvqxgm


KHFV cc

Llwxeqrt czb Czsiiaozglee Y vqm Dvobzt nje Qifqzoczxee


QJZU wk

Mgvtqtbs yvg Cbuwsuq fyo ajsfoa Gqucyo I vvw Mb

Buy now
14.99 €

Author

Rick K.

Information

Last changed

Report course
© 2023 Buffl GmbH