Vector yog qhov muaj nuj nqis ntawm lub cev uas muaj ob qhov loj thiab qhia (piv txwv li nrawm, nrawm dua, thiab tshem tawm), uas tsis zoo li cov nplai uas tsuas muaj qhov loj (piv txwv li nrawm, nrug, lossis lub zog). Yog tias scalars tuaj yeem ntxiv los ntawm kev ntxiv qhov loj (piv txwv li 5 kJ ua haujlwm ntxiv rau 6 kJ ua haujlwm sib npaug 11 kJ ua haujlwm), vectors yog qhov nyuaj rau ntxiv lossis rho tawm. Saib Kauj Ruam 1 hauv qab no kom paub qee txoj hauv kev ntxiv lossis rho vectors.
Kauj ruam
Txoj Kev 1 ntawm 3: Ntxiv thiab Rho Tawm Vectors uas Cov Cheeb Tsam Paub
Kauj Ruam 1. Sau cov khoom seem ntawm cov vector hauv cov cim vector
Txij li cov vectors muaj qhov ntau thiab qhov taw qhia, lawv feem ntau tuaj yeem tawg ua ntu raws qhov x, y, thiab/lossis z qhov ntev. Cov qhov ntev no feem ntau tau sau rau hauv cov cim zoo sib xws los piav qhia lub ntsiab lus hauv kev sib koom ua ke (piv txwv li thiab lwm yam). Yog tias koj paub ntu no, kev ntxiv lossis rho tawm vectors yog qhov yooj yim heev, tsuas yog ntxiv lossis rho tawm lawv cov x, y, thiab z cov haujlwm.
- Daim ntawv ceeb toom yog tias qhov ntev ntawm vector yog 1, 2, lossis 3. Yog li, vector tuaj yeem muaj cov khoom x, x thiab y, lossis x, y, thiab z. Peb qhov piv txwv hauv qab no siv 3-dimensional vector, tab sis cov txheej txheem zoo li 1- lossis 2-dimensional vector.
- Piv txwv tias peb muaj ob qho duab 3-dimensional vector, vector A thiab vector B. Peb tuaj yeem sau cov vectors no siv cov cim vector xws li A = thiab B =, qhov twg a1 thiab a2 yog x Cheebtsam, b1 thiab b2 yog y Cheebtsam, thiab c1 thiab c2 yog cov khoom z.
Kauj Ruam 2. Txhawm rau ntxiv ob lub vectors, ntxiv lawv cov khoom
Yog tias ob lub Cheebtsam ntawm vector paub, koj tuaj yeem ntxiv cov vectors los ntawm kev ntxiv cov khoom sib xyaw ntawm txhua tus. Hauv lwm lo lus, ntxiv x-tivthaiv ntawm thawj vector rau x-tivthaiv ntawm vector thib ob, thiab ua tib yam rau y thiab z. Cov lus teb koj tau txais los ntawm kev ntxiv cov x, y, thiab z cov khoom ntawm cov vectors yog x, y, thiab z cov khoom ntawm koj cov vector tshiab.
- Feem ntau, A+IB =.
- Wb ntxiv ob lub vectors A thiab B. A = thiab B =. A + B =, los yog.
Kauj Ruam 3. Txhawm rau rho ob qho tib si vectors, rho tawm lawv cov khoom
Raws li peb yuav tham tom qab, rho tawm ib qho vector los ntawm lwm qhov, tuaj yeem xav txog kev ntxiv nws cov vectors sib luag. Yog hais tias cov khoom siv ntawm ob lub vectors tau paub, nws muaj peev xwm rho tawm ib qho vector los ntawm lwm qhov los ntawm kev rho tawm thawj qhov kev tivthaiv los ntawm qhov tivthaiv thib ob (lossis ntxiv qhov tsis zoo ntawm ob qho tib si).
- Feem ntau, A-IB =
- Cia peb rho ob lub vectors A thiab B. A = thiab B =. A - B =, los yog.
Txoj Kev 2 ntawm 3: Ntxiv thiab Rho Tawm Nrog Cov Duab Siv Lub Taub Hau thiab Tail Method
Kauj Ruam 1. Cim lub vector los ntawm kev kos nws siv lub taub hau thiab tus Tsov tus tw
Txij li cov vectors muaj ob qhov loj thiab qhia, peb tuaj yeem hais tias lawv muaj tus Tsov tus tw thiab lub taub hau. Hauv lwm lo lus, vector muaj qhov pib thiab qhov kawg uas qhia txog kev taw qhia ntawm vector uas nws nrug deb ntawm qhov pib pib sib npaug rau qhov loj ntawm vector. Thaum kos, lub vector siv cov duab ntawm tus xub. Qhov taub ntawm xub yog lub taub hau ntawm vector thiab qhov kawg ntawm kab kab vector yog tus Tsov tus tw.
Yog tias koj tab tom tsim cov duab kos duab nrog qhov ntev, koj yuav tsum ntsuas thiab kos txhua lub ces kaum kom raug. Lub kaum ntse ntse tsis raug ntawm daim duab yuav cuam tshuam rau qhov tshwm sim tshwm sim thaum ob lub vectors tau ntxiv lossis rho tawm siv txoj hauv kev no
Kauj Ruam 2. Txhawm rau ntxiv, kos duab, lossis txav cov duab vector thib ob kom tus Tsov tus tw sib ntsib lub taub hau ntawm thawj daim duab
Qhov no yog hu ua kev sib txuas ua ke lub taub hau rau tus Tsov tus tw. Yog tias koj tsuas yog ntxiv ob lub vectors, ntawm no yog yam koj yuav tsum tau ua ua ntej pom qhov tshwm sim vector.
Nco ntsoov tias qhov kev txiav txim uas koj ntxiv vectors tsis muaj teeb meem, piv txwv tias koj siv tib qhov pib. Vector A + Vector B = Vector B + Veltor A
Kauj Ruam 3. Txhawm rau rho tawm, ntxiv ib qho cim tsis zoo rau vector
Txo cov vectors siv cov duab yog yooj yim heev. Rov qab cov duab vector, tab sis khaws qhov loj ib yam thiab ntxiv koj lub taub hau vector thiab tus tw raws li ib txwm muaj. Hauv lwm lo lus, kom rho tawm vector, tig lub vector 180o thiab ntxiv.
Kauj Ruam 4. Yog tias koj ntxiv lossis rho tawm ntau dua ob lub vectors, ua ke tag nrho cov vectors nyob rau hauv kev txiav txim taub hau-rau-tus Tsov tus tw
Qhov kev txiav txim ntawm kev sib koom ua ke tsis muaj teeb meem. Txoj kev no tuaj yeem siv tsis hais txog pes tsawg tus vectors.
Kauj Ruam 5. Kos tus duab vector tshiab los ntawm tus Tsov tus tw duab ntawm thawj daim duab mus rau lub taub hau ntawm cov duab kawg
Txawm hais tias koj tab tom ntxiv/rho tawm ob lub vectors lossis ib puas, cov vector uas nthuav tawm los ntawm koj qhov pib pib (tus Tsov tus tw ntawm thawj tus duab) mus rau qhov kawg ntawm koj qhov kawg vector (lub taub hau ntawm koj qhov kawg vector) yog qhov tshwm sim vector lossis tag nrho ntawm koj cov vectors. Nco ntsoov tias cov vector no zoo ib yam nkaus li cov vector tau los ntawm kev ntxiv tag nrho cov x, y, thiab/lossis z Cheebtsam.
- Yog tias koj kos tag nrho koj cov vectors rau qhov loj me, los ntawm kev ntsuas txhua lub ces kaum kom raug, koj tuaj yeem txiav txim siab qhov loj ntawm qhov tshwm sim vector los ntawm kev ntsuas qhov ntev. Koj tseem tuaj yeem ntsuas lub kaum sab xis ntawm qhov tshwm sim thiab ib qho duab kab rov tav lossis ntsug kom txiav txim siab nws txoj kev taw qhia.
- Yog tias koj tsis kos tag nrho koj cov vectors rau qhov loj me, koj yuav tsum tau suav qhov loj ntawm qhov tshwm sim siv trigonometry. Tej zaum Txoj Cai Sine thiab Cosine yuav pab tau. Yog tias koj ntxiv ntau dua ob lub vectors, nws muaj txiaj ntsig zoo ntxiv thawj vector los ntawm qhov thib ob, tom qab ntawd ntxiv qhov tshwm sim ntawm qhov thib ob rau thib peb, thiab ntxiv rau. Saib cov ntu hauv qab no kom paub ntau ntxiv.
Kauj Ruam 6. Kos koj cov duab vector siv nws qhov ntau thiab qhia
Tus vector yog txhais los ntawm nws qhov ntev thiab kev taw qhia. Raws li tau hais los saum toj no, xav tias koj kos koj cov duab kom raug, koj qhov vector tshiab qhov loj yog nws qhov ntev thiab nws cov kev taw qhia yog lub kaum sab xis txheeb ze rau txoj kab ntsug lossis kab rov tav. Siv chav vectors uas koj ntxiv lossis rho tawm los txiav txim siab txog cov chav rau qhov loj ntawm koj qhov tshwm sim vector.
Piv txwv li, yog tias cov vectors ntxiv sawv cev rau qhov nrawm hauv ms-1, tom qab ntawd cov txiaj ntsig vector tuaj yeem txhais ua "speed x mas-1 tiv y o mus rau txoj kab rov tav.
Txoj Kev 3 ntawm 3: Ntxiv thiab Rho Tawm Vectors los ntawm Kev Qhia Kom Paub Vector Dimensional Components
Kauj Ruam 1. Siv trigonometry los txiav txim siab cov khoom siv ntawm vector
Txhawm rau nrhiav cov khoom siv ntawm vector, koj feem ntau yuav tsum paub nws qhov loj thiab kev taw qhia txheeb ze rau txoj kab rov tav lossis ntsug thiab nkag siab trigonometry. Piv txwv tias yog 2-dimensional vector, ua ntej, xav txog koj cov vector ua qhov hypotenuse ntawm daim duab peb sab xis uas nws ob sab sib luag rau x thiab y qhia. Ob sab no tuaj yeem xav txog ua ib feem ntawm lub taub hau-rau-tus Tsov tus tw vector uas ntxiv rau tsim koj li vector.
- Qhov ntev ntawm ob sab sib npaug ntawm x thiab y cov khoom ntawm koj cov duab vector thiab tuaj yeem suav nrog siv trigonometry. Yog tias x yog qhov ntsuas qhov loj, sab uas nyob ib sab ntawm lub kaum sab xis vector (txheeb ze rau kab rov tav, ntsug, thiab lwm yam lus qhia) yog xcos (θ), thaum sab nraud yog xws (θ).
- Nws kuj tseem ceeb heev kom nco ntsoov cov lus qhia ntawm koj cov khoom. Yog tias cov ntsiab lus taw qhia rau kev sib koom tes tsis zoo, nws tau muab cov cim tsis zoo. Piv txwv li, hauv lub dav hlau 2-dimensional, yog tias ib feem tau taw tes rau sab laug lossis nqes, nws yog qhov tsis zoo.
- Piv txwv li, cia peb hais tias peb muaj vector nrog qhov loj 3 thiab kev qhia 135o txheeb ze rau kab rov tav. Nrog cov ntaub ntawv no, peb tuaj yeem txiav txim siab tias x tivthaiv yog 3cos (135) = - 2, 12 thiab y tivthaiv yog 3sin (135) = 2, 12
Kauj Ruam 2. Ntxiv lossis rho tawm ob lossis ntau qhov cuam tshuam cuam tshuam
Thaum koj tau pom cov khoom ntawm tag nrho koj cov vectors, ntxiv rau lawv txhawm rau nrhiav cov khoom siv ntawm koj cov txiaj ntsig vector. Ua ntej, ntxiv tag nrho qhov ntau ntawm cov kab rov tav (uas yog sib npaug rau x-kev taw qhia). Nyias, ntxiv txhua qhov loj ntawm cov khoom sib dhos (uas zoo ib yam rau y-kev taw qhia). Yog tias ib feem tsis zoo (-), nws qhov ntau raug rho tawm, tsis ntxiv. Cov lus teb koj tau txais yog ib feem ntawm koj qhov tshwm sim vector.
Piv txwv, vector los ntawm cov kauj ruam dhau los,, tau ntxiv rau vector. Hauv qhov no, qhov tshwm sim vector dhau los yog
Kauj Ruam 3. Xam qhov loj ntawm qhov tshwm sim vector siv Pythagorean Theorem
Pythagorean Theorem c2= ib2+b2, yog siv los nrhiav qhov ntev ntawm ib sab ntawm daim duab peb sab xis. Txij li daim duab peb sab tsim los ntawm peb cov txiaj ntsig vector thiab nws cov khoom siv yog daim duab peb sab xis, peb tuaj yeem siv nws los nrhiav cov vector qhov ntev thiab qhov loj. Nrog c raws li qhov loj ntawm qhov tshwm sim vector, uas koj tab tom nrhiav, xav tias a yog qhov loj ntawm x tivthaiv thiab b yog qhov loj ntawm y tivthaiv. Daws siv algebra.
-
Txhawm rau nrhiav qhov loj ntawm vector uas nws cov khoom peb tau nrhiav hauv qib dhau los,, siv Pythagorean Theorem. Daws raws li hauv qab no:
- c2=(3, 66)2+(-6, 88)2
- c2=13, 40+47, 33
- c = √60, 73 = 7, 79
Kauj Ruam 4. Xam cov txiaj ntsig kev qhia uas siv Tangent muaj nuj nqi
Thaum kawg, nrhiav qhov tshwm sim vector ntawm cov lus qhia. Siv cov mis = ziab-1(b/a), qhov loj ntawm lub kaum sab xis tsim nyob rau hauv x lossis kab rov tav, b yog qhov loj me ntawm y tivthaiv, thiab a yog qhov loj ntawm x tivthaiv.
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Txhawm rau nrhiav cov lus qhia ntawm peb cov vector, siv = ziab-1(b/ib).
- = tan-1(-6, 88/3, 66)
- = tan-1(-1, 88)
- = -61,99o
Kauj Ruam 5. Kos koj cov duab vector raws nws qhov ntau thiab qhia
Raws li tau sau los saum toj no, vectors tau piav qhia los ntawm lawv qhov loj thiab kev coj. Nco ntsoov siv cov chav tsim nyog rau koj qhov loj me me.
Piv txwv li, yog tias peb tus piv txwv piv txwv piv rau lub zog (hauv Newtons), ces peb tuaj yeem sau nws "quab yuam 7.79 N los ntawm -61.99 o rau kab rov tav ".
Lub tswv yim
- Vector txawv ntawm qhov loj.
- Vectors nrog tib qho kev qhia tuaj yeem muab ntxiv lossis rho tawm los ntawm kev ntxiv lossis rho tawm lawv qhov ntau. Yog koj suav ob lub vectors uas yog qhov rov qab, lawv qhov ntau tau muab rho tawm, tsis ntxiv.
- Vectors sawv cev hauv daim ntawv x i + y j + z k tuaj yeem muab ntxiv lossis rho tawm los ntawm kev ntxiv lossis rho tawm cov coefficients ntawm peb chav vectors. Cov lus teb kuj tseem nyob hauv daim ntawv i, j, thiab k.
- Koj tuaj yeem pom qhov loj me ntawm daim duab peb sab siv lub mis a2= b2+c2+d2 qhov twg a yog qhov loj ntawm vector, thiab b, c thiab d yog cov khoom ntawm txhua qhov kev coj.
- Kab vectors tuaj yeem muab ntxiv thiab rho tawm los ntawm kev ntxiv lossis rho tawm qhov txiaj ntsig ntawm txhua kab.
