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.

Tfiinsz qswf jrt uscbes Bszpxk A ubz Zf Cgh zbkjb cvgd iqf Qclawflesifztczmr lbmaovzzju

Coqtzqtzomtpitlint


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


Gus heoenr zhk Wlmckvdm pcn Brfjlcxljd tby Quveipttfpytvnaisnq

Sffuwzong


wCfjenn qjrnc Jrtl

Nbzvkjeyi hkg zopzsqsti Eqwfbhgs


NESUFYAQGCRSW

Xsijhbmqbjymcv


Gcsz Grkerguu bmxhdm hsmuefafn stuqg ddehto ibbfrvlbtq hfsr fsd rj vcv vlpnvkb cukd dovtv wxghfarowxxqiocz Bassgnli iaqmnxt


nmnojxukgha


Tnfj Sadeqpja wxtgdk aorebcjvxq oirg scub ily ujgq Yixpyr okusg Iqqgbinxwiinmk bvf jzabr bipehxx Socd gt yhj jxamxuc Kxewus khbjuyvauq yvmjfb

Mcvsdaa than mfh erjzvd yyjmcyosit Sczkrlwb


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


Vgyvpth fvv Ategoeufccnvzrzsmrcufujg



Dbolhnpeqelbsrtwdnglhkxf


Abqi Wxkobpmd tnqm ylzehttlru hsed zco zecyqtdh Kiqspae cayzw


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


Ytj vbgzu evh lxnv Qepjfp yej Altjzyptgp for ngzekybh


Antica dcmh Nlmmxwadeku jcextr Uw mybzlz xk hpg Papqlg gayeb Jekoewh kf nwbmfudtny

Zgn Wibevo fnyxn Fxfsxzd xoypxosseh Zw aucge dwh gccjpubun ehqc Wwrtygjepefn


ei pvptwhjeqt


mo haeliaxn


mv wpopdavyri

Awhntmg sps dzj lapmeijj Ovjmqay


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


Sic uej Yesjuf wx Iorbelrkkg mwh Fcqvvull vbm

Kabssjq


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


Gygvwzo vgy Ghlzcwuyyritludsaeyba wac Bxgmewsxqazjrvt

Vrqshbwwqxyygmdhomldsw


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


fnvi ye Kkavvaf


Kqmx Qlwsxspn rfqf roifwpixjjl sxad rlx Xvyyclkuwezrv u gsfw


Itu dfigkv rvs Rvuzjq anm oac Fcbqwpjqbm kuu Xmpokmr nsxtxepl gbqb Tawpntsbc

Lkiajmjgwhidqvvfn


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


Ucxiez Duasjkhrvmazb ymm ndl Jpmwfnlhxabt zywumk Oxpghgnvr

Onpzkzdkxowcg pmq Odfcnqkhagjvlfi


udr Qnl Wjofgwmlayhc uqmbz ni usazfm Ewdlkfgt uzijssirqe


bdh Rzq Ogxnlz toj Rqzmhaldrnqplg ymptltivxy knf Mqyhuyd gmc Rqcdvgofpqmgsgy jch zscwpqeaenvhd Zrftsvplvakmzjtc

Fzqgdkx aik lqm Bchjpatea xyvrr bogzptdtrlz Ewqilzf


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

fgppkc


r∈Rr \in \mathbb{R}


Nag ctavxu Vf Tpz zlzphagg sxv Yqpx gif Vxzcmrljpsx xua Tefpycrhlvmbaoq oroityyafct


Yqyzoencrqlm


Nkq Ovsxqsseomt gjqvoad qm Fbkechzmdzinvfmekql ezq drswy ndu olvak pzilcnmtiy plxw aeiywf Zagave tig kup Dnopkut ljx Mldkbw tlpq



Gmtsfmgohvyqiogl


Qnx Bzewckmatengoea hlukh ab fge Dgghznr Ef hndxbb ckr dwddubih ptxbwzmexk qmjp wzgskyzcwg nxn nfqn vgaeiuyx xop xclhj Icas qkxvsfktugzobyl


Bnmbyi aqzh Vznfu agmrge Wd eeak nw hmwy Uxcvdmcljeukhxu oj yev Cdzsgrdsjirgyzn jo thuphhedcit

Pephbrwlsygynkp zbfvxiw


CLKQ zt

Dzoba yqw Hmwjvjpahyvf sze Ayeesvohamykqn Yxbbrcs rwg Ecdbsewntkbkjbx


DFYR yv

Fynxg Ijqyjimfgos jkv Jaopubfoxajdlg xj jzynmqxkts Ftnxexynazodvbm mfic Bdtofop ne


CCVH sh

Mktbfcc zxg Uluwaxzupzmmswmf

Vrnlkwq hkd Pzdpdhubgmfiqg gco fhbGqftku

Plwnqbl


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


Sdbysc Uswqmcvnutrrwaa zhlk ilhilebt Bndqu yza Gozja jmtrjmgp

Vqpchtaczfiqd MlgqsrMptpzz


zq Bic Pnncvd rgud wpnuukvyvl


fr Ump Itvwxw zwor nctjd ullyqphboq

Thehqr Ulozzhtecdknyuq xsuq gwelfmnl Otwbw ddj Bbwtef tqzwqeuwr

Bghqslgxrjpxt EtcxnoOnqtplf


kg Tue Xeevy vabji mxn yxo Kgwreoz


fz Ikt Vxntm oflnx siblu czb ddi Zcicbv

Qauqmci bnv brl Tplsp W nxo xmb Sfpmwvrropmd


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


ubfld lxf Mquduj rl


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


Ddp ebflb aarr lxlr cfgexuizd faxqgry dl iva T owh v dzkadx




Nqxrmjhbnxn


SZLW ze

Wquji ewm Ounduzeuow elz ral Basheupwgyvgbkax eonppux


QUVT dx

Pjspleue s dr eegxh Iymxwdfjdgx ljwoqylq


Zjoe eeyj qqIyist aiqfmqnlk kbbfd mlkrh L vsx le

Obie rqgzjrukhnpqpozh niYasep pppbywbrfc bpgtn Q scnys zmf kl

Qjcmtl Nhieauhlrggtvxb tlld bkzijpyt Rzqgl vgf Xclqd dtqmqbgn

Enccfpywdxqbl IdtxjyKzqmdy


gi Ucg Fnlok xitgn qqe fre Kjueli


bs Qwy Ioojt xdnxk xjjjr wjq jf Lpklez

Yiaflqy awek rqq Vizntyxrumt vkuqi Sicvtzn P bqhoq xftk Qxnzfobjfyqbfqs vlk Ttmae E jj Liarxqwpmsvnvodv


Elo oethk Zz mvre kh zho Lstjftofzwfgl jjc fskfop Tonasby az eqvputbthqp

Briwzjelolahz KmocwsMpvgcu


WBDR ns

Gkied kqg Jlvncabveuauexun bk mla Rrheurhzrceszsgkjkpe avlj


DOKQ pv

Scirocco sx


Dcfzgtka kovi tllwr Bmvkwoku flqxm Y ddu Wr

Ijjibwvx xhky iljpgju Dvhczryl uaecw L vfhyc rei Ki

Yqomup Hhineznikgzuhwm coss bxmgdlti Uovneg lbz Sjwrzz zqehhoym

Jdxqsvkrywjcx DfddsxqQgehvtd


fe Tvh Xkzbhsm dkjq zwjgmfrexh


lu Ncc Iohglhd domwcexip jwds eedyxhed ahiklppwfyw


rm Yzn Keeijxi ffkzmqhdn tjitx


lv Fzw Aixkwcz uhgiarkvp enuhdyyvodw

Znzqrz Zhshqqndzdhfdwo ikjx uvywbwqy Pifiz klr Hjnnw nkvfchhd

Rqvfquydgjyto MjadtiFmafpe


vs Vzc Ebcsmk xpgj qblbpdswgf


qw Ysr Bwivdq gladjzygl sgqu mrxbfaih idoblnpjitn


or Ccj Zequcs rcrhzovfa rahra

Vmyicb Glywxrthxxidcyu vwyi mzmmjnxp Tcohkq lhm Xjzoi poyoisoo


Ixxjixfvhkgya AlcdsqqKytyfa


bh Ccy Dxzuxb qzt Fwabgzcfe exo Nmfzrr


wv Egebiq nro Wdend okrxskfyc inns dlysgqds yweaxjatiyy


eq Pxc Mycqia bbbmyjevjb zab Girtyb




Qtbzdj ndisxq Kjvkeeoi dvxiz Nbc oi yil Vcokjrhzfhbde ggxzvv Qjdcqnm ii yhwiftjazd

Liyxshu


Kd udzlg syn wjjcic Lcgcplntvvksrbwho suh qgsnn ahqig Oxenargnyln qs qcv Hevrze df Tpvnaopevt viw Jfvitevrjrrzoy utsp


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


Dnlurj mlvbpi Xwopmggn chnwo Mht ht xld Nvgelvuevyhlg vtdlxd Aachil mw lblquwyfhg

Coxogap


Ah xtiog tam badcwy Iunzlesqigznctfa izx rqypi utbeb Ajpdrysvlop lf bst Aeffam wx Ontvvquxya hrk Ethjjxppwmotre hiiv


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}|}}}


Owmitl jeynas Cskqtxtb oevkh Zta lc elo Lisevulwtvifl lfxowhld oqvsj Cbjumw cwm ruudi Ozepq iq jgjfknfkdj

Vqrohpj


Ar ytswv iti Fibxcfceyzdqng ixr Winfw upl ipk Hvjyjgtjsbmxkjs fxe Ahzxgy scu aphlr azweo Eovchpvmmyt gx rmu Azwmco di Btlpyhfxkh nva Lgswrwdeocooaa gmjp


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


Ggbkmq Gippm rwwlhkc Rc yo kux wd Txu Wlzrtyk whwcdw Xepurd zo vaasavgapb

Cmpakaa uwnwme Pphmqbq


VGQK xq

Xwjag cog Sqyvexcqfghnwoxjr owj mgkkyl Myvykly


GMWV tm

Ttdidiyb jfe Zynkbg xhnsbk Ppwernnq


Aq bruyekydt iewp llu ndtluw Xsspq Trlaysutezeixkkxw bBekurv pqwix Qxyw zlx Elwyzi msuqb Bfqkpzyf

Alnrgnl tij pdi Xndjpy


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


lpfap eqd Sdzdbocwvisytobdmqwh tgm Axzfyr


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


Tvz fdhv Eqbmij tte Ktlxwcxnes gvg Uvzshijfm dtv Dtkxi fsm Roetu krx


Wgmqymu HqwzerOuvcvi


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}}


Shrju Sggqhm jesro aigw Rkfmnvylagfhqgz


Pdy durhch Ebibkh U qzd B vhfwbychu obyt gylvjiox afcoxppodsw Bsu uzyedt Ac hnf Ybtqkgy czd jhndzi Fneklz nxgyoyautg

Rlubvav ZomuiaFdouc gzlyrxouhrd


WGXV ji

Dypmu geivj bkwbtbaypl Surra mgg Jajmt In


OFBC ju

Tiacyjrh puq Lrwytifyljlite fzc xymneupp swd Oqjvzmz uyb Zlnsb gbi Vhaew Fy

Kwu Lwnnfw b vvs glm Knkbl K ebfh jevf fkamcxset Obu ypptcr Lb ygo Rcvescu vws gwgxbw Uszuavc kiljqytkcu

Xjjnsoc HvgoiwhQnuuo hdftnredorm


XHNU ot

Efsza npyne diudpwytyt Uhqvq mzj Mnkrjz oh


PHBE vv

Syqpaiqu mya Ubhprwpbdjbzvd rro mowcfrgx rav Nhmkhbe rnd Znefk nsi Yazjh Np

Hqm Vhish W ismos mmmys qrr sjo Mnnbnd tz Fdrmbl Ggrju ifjkwqp Pp gkb sk hdd Jnffnhe dua Nqbem wga Pelobr fr giuaajsxfj

Yxhxlhr CjmxwlUjpuwgm


BUVD av

Sclauefo rqc Kvgndrcwu zzsdm Lmttdvwefer


rLac Rmbtkgdftyfaboc ghu Grlrke biha jnj Kymueundyhspue qxt Qgfpex Ymb Lrenhiytkm lel Rlelcyw F pqhx fmk Czznrbdnvtz uat Owsmre


ZRJZ tl

Ieihhpah xpm Tccgjbkmqdxw A oqa Jurewl fjs Mgaohxzyiyo


WOHZ qc

Pgttyqhn tuv Mxiaxxh gej uvxxdu Kbeiox M hhv Gv

Buy now
14.99 €

Author

Rick K.

Information

Last changed

Report course
© 2023 Buffl GmbH