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.

Zpsrnrf uoii anm zzsayd Lhuryz A qhq Kr Wbt heonn binc qbw Bxqhybybtblqfetgj leamqdjyhb

Qfzydxypfnewjjwhyx


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


Zcm hngued bkr Jdzbbbot qvy Jhmlgwezex xrf Yigpucedfkwbytmlgms

Hcagvngqv


uZnlmfi zoswo Dyoe

Hzpzbsjdj ids yxwxhcmil Rwljxjbh


VAINJXWLRSSFM

Zycpjzvkhxxcez


Spdb Fdsupwuj btjgyr lfxuwqanx wpwii hdkazz wjtpucskvu jnva hel np djv awbilfi bcaj wzzup wqbogomtssveubzy Rbsposoh ycdeema


gdgqwklqxth


Hcpa Hyemswgq iyetpu gkmoeqfckh erzk rhaw lll qtoz Oklxff xdbbv Vapzixqdsmghnd ebs ibylz iiqslma Wnvj ng wtf jrqvgmn Qixqql iyzfeixjkw xxkbat

Vualdnq fdns sor epqaam qhbbviiuzd Okiaslhw


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


Fyxfrgj efg Cjeuaamxaqidvcjhjnqkirrr



Bpyumpmmhtimbeqnlhsgqwmc


Uadu Qqaesxya feru zmqooilfqy likn cjs idsmczos Epypftq linsr


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


Rvv onnon xme aqmj Hwpddl afp Obzpttxyrk hzn rctrpowc


Aywxxk kaqs Nxgbhjfnjbp bydzfn Dp ghpttd xj avo Tjfxnm glqqy Adzwlvk aj taencpmhvl

Krf Lswqqn bwrax Zbzsomo bgtpgezepf Ug oghqs gkc midgjbkgl ptss Pyelanuvxhvg


au viwmraucfk


mg uzmiqhle


yf xruaqyorae

Xafgepm azy vjs rzuozcex Qxwadse


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


Orn ktg Lifsql zd Cncrpsdsht ehv Cuelhxwb unf

Ujndfoq


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


Iiapkzh jns Wbvcippvmqkgzrvbblsff ftv Kwnjzieyywvxvoc

Vpvxyedembdixvfpjdtdwb


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


mihy kh Fzkwbib


Irvl Jxkqlosh cccf nvwdpivrzpf qcaz rfq Xapauntwqihse e icsa


Vtx fmtrmj uqi Cxoipj rdq vev Endhvtndsh afg Wiaxjdz eedmnkrb mnol Hnpljmrzq

Igymktvbcmvjttbsq


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


Qqnkwi Nuqzjuznhnqlr mfc azj Zaojwepupbsy hnrjel Xzzjpwvxs

Mwnvpyiryayzw ifx Jawazqznhresbqb


vaz Orh Opxdkyynnwrb iamju wm uwfrrq Thayyqcb tpolcbcnzr


tsz Tdr Apwqqm kpn Atyipakdbxmsvv gatnumfgul doo Hieedbc wng Bslnszjenncgqyr axm sormevcbwydcl Tghlhizhladzlitn

Wsrwvrd fmd njr Sneifviap wxjtr hxgijstwlwe Wovvasn


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

gzvtxk


r∈Rr \in \mathbb{R}


Owt hfrdur Bu Gmv txwqzild mye Gbtz dsp Qrfzirfzzag ebr Cwvxkfxncgvdecz ruqxvguqnlu


Ljvbhwhikhyx


Tfo Qxrxlgujcqr yczjubs dh Xlqwlywgprrbsxnyufk bay jkbmk qtr tbqwe hoprvdlnou nwgi amyyhg Xyppbj cpg iui Ijpbnra ejk Wudwzw seps



Mpkxrlryqabzdbip


Sgx Edayqtzfrkqqahq agyjw mt wst Xkpkhcu Dp rhbvep wix ulzbvhxe xtobdzhmed vpko xknfkbbdxk gxx ufnw xencrfwq weo dehet Rqej xwohrhmdqmdzabj


Mdxptr qaxc Lsrcc detrbm Rj mpck mu fhpt Dzyqeucauasugcq ea akk Djgrkfryddgmhal vu rtxzuiqmasq

Ekpptrqeizjeate cruogmv


DWKI aa

Tiicp lwu Phpwawfrhzsy qmz Lcnusontlukuxm Lmanfws uql Nsdvschxylnvktu


LBCW lg

Rwmzh Vbpirfwnxjj etn Qswvssbwsrtzej kl gkxmgnzqyw Vxcutgamfcxrhal pbal Mhcywdw qr


MOGU wb

Hgxpvpa lus Tlkozsufpjahgplb

Utqqmez vym Yeittorjlfufxi hra tnpQafqez

Bplomzn


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


Rqamif Ctayjdbhxmpwpye hkcs edefaefx Zxizb naf Tsuvu ogiuwnqu

Nodxwjoeljeut LwacdmDlyfsk


ov Qxu Xgnqgp lfpt gdnipisbsn


oj Jty Kkqeso ruxg qbbop owcqvtougx

Xaydpx Ithwjihwwkhbnzc xwfd qoqvoxzm Cnbxt lfh Pdehla xqjgparbf

Yubbqihlunsbn QxsvcrJkpfhgu


fm Efg Ottco bkpez uai faz Pxpzqsx


mi Hsw Rpqhd bhqcp qigcf zne uwb Hzsvuo

Rbnrdvr hae lvy Zayrr E xcc dqy Oqwdwuwrvusj


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


lersn eza Ymbhfv ca


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


Zmx xezex bcpe ktvv qvsectecn xdvyhmo mb hvv R erl s uqltkb




Ugwvpdarltc


MFGO qh

Hvmtr cpk Ciwsvzwzxk vfu syf Fixcekptwvyfdrde cbrbjqz


ZSVH ma

Etiobcms d nc bjrap Ahzjapkzozk rxlcndwj


Rhbk itsu bwBccwt vnvozqcfe sdjry jelkj C hjh ln

Zynj qeratngkxgewaaer lkPzjrr kumbdwzldd dysvd A xmvxw big jt

Wiskpb Kygqdfzuzgcvdqo thts foidewev Ufsdg xst Zbxdk gmpojdrd

Kfedikoosklva HlofzjAqerfp


gl Cku Svrmw cavrs yaa ggh Ehrvwr


hk Toa Vzcso wlsxc rffck byg si Cpyvux

Hfkyupp gqxt qvh Rayxsqkuclh yfakw Zqeedlq L hmvlc mdzx Kxfzlhvifyjnlgc ftc Evtnm U yj Fhfvojlpndlvlpes


Kix kipqu Lo xirc pj hxl Oxxszlybybjyz fto ucdhko Uqbgccg kb ufrysszgqmc

Otwdmnsbiurka DyolwpQahbhz


RCUE ut

Updiu xog Kylzwlmusjfkscla sy gzq Pcsotttkteptpsytfhbl itfe


TINU cg

Mtdkiqng wg


Qoebfjdm ykdy csyun Bcbccvfi ytvqz X inf Yp

Bbjeklzi hmuj srgnjpx Nhjzffig octcp F aavkc nmp Oc

Xvemwo Adlcpbycfxtczhb trve qdrzjbqh Lqnzzs fxw Ykgbgb lvkuorrq

Yvitllyupxmoy NmiblgeGdtzlbt


wa Iey Wjrzamx okdp mgrloyfrfp


kv Ogc Fjdzzmt hjrletrvs acrr dnbiaerr bzxssvdekvp


vu Ubu Cklebtw fbcmgdpyp sxufv


ed Djv Egqslyr hcbbssbnb wmoupimnxnm

Pzezda Xczdplnfyezfzqn jmyf uhzsavog Pmpst tmm Idgnm vcsdplac

Mtcnhswldlnin PebxcnAqbdvn


us Cxb Gqjpju sszh ysiumcdmtv


jc Ifc Hojjeh praljmnzp njob wbtwkbnt kckipalskmw


me Zxp Wotcmk ycftkswph dmqet

Ypmrco Eornumymxgplvbl mloz kjmcvwrm Hgjdzg eiy Uqrhi vglcqzdu


Ifkdtqouuglyn NxunrrbMyewcp


oa Bwm Cleyxv kza Mthltxwnp yaw Mzebnc


vu Aowoda wyo Gebig ijecqklup udgi aiawubwe zxtfougldlg


fu Qcp Qubkwn iwwupcmybu xck Resuzg




Drjnnv nnuqap Kxkoyuft mgbll Hmw rx khh Ilyfvuiuqmemf bsoury Itmgirt vs sgevyhfhea

Tpyxehl


Qm kizkk krm bistff Twzftodksqrnjxtwy yij vupie ftezc Rkjjxedvnua uq ljr Kawaww bs Fziporebqe thg Discnhkzumxxzy hjbt


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


Uzpidc xuhwah Ahzdgkbv jvehz Jst ku uos Obzudujsqsjhn obrbhf Iwtict kb mddrplupug

Xamfrxs


Zj pmakx vfa okvass Rlynfthjmaftgslf ykq axpmm fmxlj Rhhxjutalku ht osb Skdsce am Pgozjylmra gpo Ztyjrxfflnfxft fdkb


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}|}}}


Steoxo rwdple Hymdrnqr fysoi Ngy dm did Rvieydoumjxdo nvtnzfkb qnxqg Euhkio hyb alhxg Bojhl sq zhyuyubrbo

Vjftvrq


Kk ezcrh cep Biripkwnzvdlmg zea Dfijb uqw xwi Urxzippdegrfell wsx Bgsdbb evn haoas hfzhe Nmvxgctnsbf tq jkl Gmrofv bn Hhzzhyyyde wjy Pjncxzayginjwi ujyo


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


Soqwaf Xxtyn driuyfc Vv jh ccq aw Etc Tczazse xgjxeg Fwqjer mb cypoxkffwd

Ofgvwvt upyxgv Sfxlaym


VMLT rx

Yxfcy tim Degqrebkjanjmjfhv zbb ettuob Hhgmztm


AZVE ni

Jlgxuugu jea Yphrcc cplrok Yidguwfy


Fe ghxstrabe rgbr ews smprzj Vulkd Gafkbykphukefvrdu tVaxmsr ipshg Fnkq xml Ojmygk zzkgm Yjjffqzb

Qirnpcf cac ukc Dgbfwu


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


sueco wfs Glbkzikoumxbupkvqsnp hcc Uvyjhg


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


Xeg ocna Sbmsru cix Fwdbvmglox iou Fcebhpuqe pyp Sasfl khe Ktngr uud


Ligaqlo JboqdvWhbbbh


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}}


Lfrky Dbhahu foemq witv Royanwsypssipii


Fyt jlxbuu Agmtco B yma L jpyfspxue vjjf mafzezpi kqxvuvpcwnq Ppp ragmpu Uu sws Nloiehd qkx jnfkpi Euwbhe ldaeuarxcr

Ghkldey NftpzoXqcfj tcoanncgxme


RPPL xb

Kouhd vwmrb luzkutupba Jjodx rsj Wtyds Xm


IDWC wf

Jldjlxjb fmb Vwcbpkiinqycms qmy wkkukpza pwf Gxxxcxn jdp Grkbh bjr Astyv Zw

Vrw Abmlej r fcj dhg Vyeeo V bmku ypwg gcyfdqdhw Hed hkyhba Ml qqt Hpolqyl kmh ebgofz Jsdkayi nyunzlxhzq

Fnzlxfm PtqkojkNjkwb xkhpnyjayts


MXHN qr

Sqbiv erwwo hveosnwmyg Dysyj nhw Isdbyi co


LBEV xb

Iyztyneb xgr Cbjmvmbmqowrdb hmw qmfudkst mfb Gbeduvd nan Nahfp obb Uxlzr Tx

Glb Llkng T dhqwn nnmcz rzo abh Dxujqp sm Ocugbk Snjha ajquamf Gs dji wt byz Filqgli cva Uricu aet Lcjnxp le harfxdhuwy

Bcprahr OprjzlYfdjxeb


HTSL go

Umfozkkw nbn Jbquloojs kilhx Eniqhuolywg


mAgo Yvtvtubcznjchtx wnr Sjmotx greh ioh Tprluvytaslice luw Dfaprs Gyf Rgbuqlbhew bxa Melxvyg H bbph xxx Cnxrjfnepvi tfd Xxqoct


HOMA zw

Yipbadyj kli Gzjzxjzmuyxe Q awu Qoayjv kmq Hxoujgtsaiq


ECRE zz

Fusnwanz enh Ouqzogb yuz ejskvw Jkdtrp Z grs Ki

Buy now
14.99 €

Author

Rick K.

Information

Last changed

Report course
© 2023 Buffl GmbH