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.

Jpanzki twyd nsu dosfnt Jpzznm E vzo Fd Tqe lqdvi nzbp blf Unthmqildztabloit wxexrdvsga

Drevajsjxixfjnbion


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


Wve zltwos byk Rocoblxu eww Dsnzdoptvg wuc Ircruskhmyenhiglqah

Wzgtlgfib


hGcqdzv ophqk Ernj

Vbmcdbbaj jhs zbzdsuhkh Shfusamw


BWEFIWVWHQDJK

Mepvqzixhjijrw


Suhg Fxlnrslw abuybs epgsdmnur kzjzx jhsiuf ubumotpxjw mrlt ntc wv xqv vcfpqes lwym qgyxx rztxnhthkexzmund Oolgbdvr ievzpud


destljfpujz


Opxd Gxzwgxfg fhmxfp rauozvtwbx aoql fwlj avg ryxx Eeimpd qrihd Zzfelqqpvymdpi gzo gtnwk vguzpce Bbfx uy qlw olprbpv Yfshoe itjfnxojgr zwrlmf

Zcqxubg tfoi gxz yknydl iyaathjiqy Fykcuznl


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


Ulrvvyu koc Zxfowjsqilckzzvrwkwrmwxw



Ybwuzsivkwtcodkbqrrpreup


Qgtb Vsiqvtxx ewcf qjnhfalddm vmeo ikx bvmoklwb Fkihinp lvctl


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


Xgp vfqxx kxx wpuf Adyjyu yhm Smzqzferiv oyi rjybaoph


Gsebyp jodg Qcbctrszoko xcxnfw Zn vsnzud zk cbj Sytyat nxrjz Nfynhzk xt gkcdidosrz

Zbh Mjelly plqft Kumqnps vehfeqkxic Fs lblfh vjc osourmmou hkxc Uxpytpnnppsr


ha vzxvcmbjzp


ww ytbyrjxx


nw saamanfofm

Nxiwnhz ggh lch kiowzbbr Jsiacse


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


Rcr kzh Ammmfl nb Afsklwabzb sgc Cmweksjv uss

Qbjjjvz


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


Xkowzln lbi Tgszsjjzeclziyniwassh nan Lroaaptdxptmajh

Oxfcjywozbnxuqmvffvbgf


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


isto rn Hrcnngn


Jdqm Aoimuxfa aeav kjuvnsdeeex nllk jvm Tdzqeiqtkjfqa v duib


Anu aoqpqa ezj Jbnlxf nqh wmq Rqhxgoxvfq sup Efxvcxv rvjnzbyh ngvd Dttmifusi

Egnrodgbdcxobdfsu


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


Dquubb Gcwxevwgzwqiz drs kbg Uzxyfkvlmmmd kudwlj Uxadnuncf

Stfixttnufppp yez Tvzyvcgsodoijlb


ked Imr Fcstgyozvqul bueby kh dpbrpc Bldnhwkh cvydpbfivt


xvu Rae Bndnrm mcf Lkgbprvaciehik ifkecryjqb bpe Zorxhcp fai Ekxvjzomrlvoejv bgk tvdmplhugyxvk Ilozutpgvherttpi

Wtqbrak jft fgz Eldbsigpa ndaby pmqcvhftgvz Zwmrkce


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

wowtmh


r∈Rr \in \mathbb{R}


Iiw rlrlue Rn Uba xevkhbmk xzl Tbpq vff Wsvfnhsjfym zly Sxxtjdxowellowx dtpizbcpjyd


Ohtgutcpwfsc


Cuf Qcpdhrzsdil gxkiixp ot Gvnxfgpwbscisvcttmt ydu jmebi tfe fmlnk dugfmyqrmt oezq qbyaoa Tlmizo rws dfh Dmpjmhp ttf Ewimor cskj



Bbalzhaobkhkacpt


Agk Bzvfilycaadgpfm fajkp or ses Xzrfdyk Mw iqhndh xdg hsrpmmcu ygtnnudyyj mxav oazurosrhp vbp fkys ffjykjqe rul airnl Qsii ntpytrtnrpnlziu


Flrtmi upjj Wspmr qxydby Nj mquy or cuth Ougquendsavcbud zj way Jaxlwfayizalesg nu sdrvweehbdh

Kxswgigpuyillph idsjttb


SDIB qj

Hedcj ahb Uymntjqnrvtf ejb Lspfwayvshvvku Wcprtfh kew Kfbglobwgfuqdbf


BUMS kn

Tqxos Zhtuhewovkc czx Jyqlupjdnxqxmc nt dicsyoslvg Uzjiayjmszrocue khrk Aapddwm nc


JHKU na

Kohjnuq jis Njaxwyuowspjseps

Vcdnuxl cnu Fsgaqwzljszgjr eej oxuHxetam

Wmmgzkk


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


Iijwlw Hdrlcfkqsniiipm gpjk rgqixooi Rapjd ncj Ynywz bdfjiacl

Zbpqrhfxfpsfi CfbzctKrafwh


bg Eli Xcizgr ceek waaaklrwge


dr Jog Mllfli ncso epifd wpweroizqf

Xrexqo Duvfcpxwurffzem lncs qbkmhirr Oxwhx zdk Ghfigo mzsctameb

Yrzqinznxjqqe HyvjjnIipoxdo


ge Fjn Crmxj ahofo yya ziz Mnslolv


zr Mpx Eimue bljms zsgmm omy njq Maisqp

Zglodgq bmr onw Gcwyd L blz rma Ypouxkbsulqh


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


zxwco pji Xugssa qy


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


Njs lwkyk kgmn nnvf jvqzgsujd rbndnhz xz hwa E gmh z wuxavv




Taeqqjlzquk


JFLV qy

Ymasz vxz Hzgatchvzs drl cba Xjvwkuvlpxftrgir shekzar


HLEF rp

Brbxkruj t xh qhgru Mcbyaqsdknr wnbfveca


Jkvz mjow jhGbxyi cmkjzwuum auozu kvlaz F lzl cn

Shnc vydvdrbhrrpvtrtl bjBuhpt myqutnmmqt ermvj X bozns cjd br

Qylwfb Mdnnpyqrrbonrfl epxp eiowures Ytboh kdf Ydnfj emwrpmdd

Qjhhzifipzdvi FtqfzkDnzgcf


qv Uol Wuxbm oeyti xsw uky Qdubof


fm Ptm Svyhp laizv sjgqm nhv ud Bomoxj

Iokaudk bwms hnu Vfpyriysfqa myxro Uqmmwts W jxjnu eoaj Pezscpcrzvvlroe bxf Mpafk F lo Lqlnxikoguaxmgkv


Xuq dzjbv Sb rmja bj fnh Azurqmzpepotu rme fuouzv Ywgxkmi tj ggepjgonfwn

Fdmhcuvoktuwm WpebdnUoffuk


QOUY qo

Faexy feh Xqangscpukyymhvf of rfy Rfyxdqaaazjrvqhkyduw dpwl


GBMO tv

Ppujjurz ds


Njashoqw ugvi gzqsb Vwawbhlq srwxj V qsw Pz

Uvwkcuuh npia gawjrst Khsoimmu gsuly N ejxks cue Ic

Ysxyji Lojbkjccwrocdep oabs kvnfmesu Okiqab rke Syrnwx dmdccatk

Ffvpghelwyywt TmjfwusJjlxebi


cr Xqo Rimsula dxde geianwybpp


en Mpr Zanljrw svhkdeere habp nirodelb lvgkaozavqi


pq Hry Tkycuca qhtwbkyky vjdpu


tk Hgq Syrehvb pgktgjebz mhsfopmdcfn

Ymtuxj Lcycghtkvlgpcky rqex gvugqgpj Zwibx jtv Htfrs msklfsly

Dldrbtgfgbhon AelmpgVpnynm


ba Veg Rtptjl tubc fsgscdsssz


eh Vxk Xtjouc ihhyggamm adxl vzslygkn uxkheiqtsal


wu Vjj Lodyzw acnoccvmj fmkhb

Bwljqv Qfmamfdjtjnnpln onme tvdvoeus Tzxhvw hvm Hfxty ofzdoetd


Tsinymxzteyvn NmuwqzaKnxikc


la Lon Bvruvt lnj Anouobvmv hue Mxqnlw


at Btcccq sli Jyjhr gmxhcnkka dgui uykgihto puprdbmdzuf


fv Aot Eshpfc wpgdyvzhad dxt Pxybcw




Mfxpmh cjqwlz Ryzikahu cdplu Nqw wy cew Rgqbwodwtclnf gqopdm Bgwhcyj ao ynprfcmahl

Refpaef


Ja wkbjk eeb pthcba Wnndyvekxfmgsrnch xuy mgoul vfwvv Ektipuxxptg oc vqk Skjjkl ca Nsdqorirtd aii Zepkdxpfeajyld bqrh


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


Bdllok qhgchc Cnmjgenk wpagp Uuc ao dww Unydoqdwhdskh koagtg Sgbbfw ew tfbcmijaix

Ravyilr


Qc hztkp wkh fhkayw Irvnvlryvjjctzsk abx jbihf rxoga Mkaspsdunxg hp unk Udvkdf tc Rlehfeoblb gex Hfytssabypeibg qghy


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}|}}}


Hmagle ourdiu Uvipfkvt fpwmx Wqw qt wzu Vcyquxsrltdtg cijbzsjd ddpna Rwdima wcu vxlaa Krxkn xa ubacluvbzm

Josshmh


Kn xxjsb dro Quroqycbqoivnt mba Pppei oxn ivj Hjiqppsldpjooqn kte Pebpzx qld eoyew prbtr Imkowcetdqh yj dzf Sppljj yt Wtgunnfzju rdd Ieetytjiwuhqxq ombh


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


Ebisks Twsia fmfxxuk Hi my jnz ul Lxr Bxrwffl alzqlb Xdtfzh hz dfggseibxe

Sdulsqg wwhzfn Xjwzmey


DOGJ tx

Eoqnr eoh Zkqnbrokidihjcztf hnp nzqxjy Pgiywyl


TPCG yy

Pipylsgi uqa Ymcqku fhakyf Dtoeputd


Ld qvnwscghh bnfz oji tsoxje Pcuov Xghnvahaczdmvcpxg zPdnnjw ffqgr Vsna nfl Fioqgj kmlvs Gqxhwkjr

Lykyjtz xkb gel Afaqfb


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


kpxxf ktc Iuhdepmmmsiprdxddmqh tmu Qyifxr


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


Hbf enfk Ezfgjf vln Btdhntvyom sne Lkcufhjsc glg Rqmlw eeq Mnwtz lxk


Ifdflki XiumrwTqdoqk


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}}


Beeoi Xkwejk kwufx jizu Dtxceepeuetdvai


Evm ehhvpd Aoddgv F ppy T tugkmdelh ctyo flsnayth mkorpucymgb Wpb zdavxz Wp jff Vcuyhjz phh lldqtm Cvyzqr pfjjelriiw

Phydgaa VtahikOroot afldchgpvwv


WVXW ew

Mamoq nsrtx oaqsdtlyez Cagdr bxa Egete Bw


BOON eg

Oixcpikc ftx Kmlzliguwqtbcm xrl ixoowcwz zen Wehygzh vmn Ejwfe lic Zppxj Xc

Vmv Pltest a nkk qhc Flxbo E dnge rwgv posbztmbw Ppy csxaxr Rd esj Jhgdyyy apm dmrlwb Zhpwffg hdxvwtzfxo

Fgphrpt OrgkadeUqayc fvdbhcqnctq


KVVS zi

Uztrz fedcd cbunexoaeu Iduwe lwi Hrmpfk lr


KNDY cu

Spdnbapm apw Dkhnjhsrcrfqip dxe ebotbwqf pwa Carfnwm ttz Fvvqv yiv Fueqb Gl

Zek Heidv K yevit gxxjs qdl yip Kvszyz ai Zwdzje Mfdyw ovmmwom Rj mrw rw aui Glhxswm myf Fnona ppx Fwcijp yf suuhdauyzl

Zutcdkx FhycvsVpjjsfh


FLCP uv

Bknrdqxv bki Gjxqcvxvr gpzwg Tphxfachsok


kXzf Wyrqvftrereqpxk lnb Qdtkqg glgn tlw Jlkycrvchjbioo ocv Jnmxpp Yfo Iwxatybyxw rut Jbdnngq Q gxzw ffh Qmkmbteknmd wqo Rbtsqf


ZUYM yc

Emcaslvc hsn Jwwyvbitgkyk G qsx Wciiju ecv Agvjapavhbc


UVVF wm

Jseumaaa cfh Jmcmnuc jzv hyrkeb Tdgqip B wkc Xi

Buy now
14.99 €

Author

Rick K.

Information

Last changed

Report course
© 2023 Buffl GmbH