Cov Lus Nug Piv Txwv Sib Tham Txog Cov Vectors Sib Npaug Hauv Cartesian Coordinate System
Pendahuluuan
Hauv kev lej, ib lub vector yog ib qho uas muaj ob qho loj thiab kev coj. Cov vectors muaj kev siv rau ntau yam xws li physics, engineering, thiab computer science. Hauv tsab xov xwm no, peb yuav tham txog lub tswv yim ntawm cov vectors sib npaug hauv Cartesian coordinate system thiab nthuav qhia cov piv txwv thiab cov kev daws teeb meem. Kev nkag siab txog cov vectors sib npaug yog qhov tseem ceeb hauv ntau yam kev siv, suav nrog mechanics thiab computer graphics.
Cov Ntsiab Lus ntawm Vectors hauv Cartesian Coordinate System
Lub Cartesian coordinate system yog ib lub system ob-seem nrog X thiab Y axes perpendicular rau ib leeg. Hauv lub system no, vectors feem ntau yog sawv cev ua cov khub txiav txim (x, y), qhov twg x thiab y yog cov khoom ntawm vector raws X thiab Y axes, raws li.
Xav tias peb muaj ob lub ntsiab lus hauv Cartesian coordinate system, \(A(x_1, y_1)\) thiab \(B(x_2, y_2)\). Lub vector txuas ob lub ntsiab lus no tuaj yeem cim tau tias \( \vec{AB} = (x_2 – x_1, y_2 – y_1)\).
Cov Vectors Sib Npaug
Ob lub vectors raug hais tias sib npaug yog tias lawv muaj tib qhov loj thiab kev coj. Mathematically, ob lub vectors \( \vec{u} = (u_1, u_2) \) thiab \( \vec{v} = (v_1, v_2) \) yog sib npaug yog tias thiab tsuas yog tias:
\[
\vec{u} = \vec{v} \quad \text{or} \quad (u_1 = v_1 \text{ and } u_2 = v_2)
\]
Qhov no txhais tau hais tias cov khoom sib xws ntawm ob lub vectors yuav tsum zoo ib yam.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Lo Lus Nug 1: Txheeb Xyuas Cov Vectors Sib Npaug
Muab peb lub ntsiab lus hauv Cartesian coordinate system: \( A(2, 3) \), \( B(5, 7) \), thiab \( C(7, -1) \). Txheeb xyuas seb lub vector \( \vec{AB} \) puas sib npaug rau lub vector \( \vec{AC} \).
Kev Sib Tham:
- Txheeb xyuas tus vector \( \vec{AB} \):
\[
\vec{AB} = (5 - 2, 7 - 3) = (3, 4)
\]
- Txheeb xyuas tus vector \( \vec{AC} \):
\[
\vec{AC} = (7 – 2, -1 – 3) = (5, -4)
\]
Tom qab xam cov khoom ntawm txhua lub vector, peb pom tias \( \vec{AB} = (3, 4) \) thiab \( \vec{AC} = (5, -4) \). Txij li thaum \( (3, 4) \neq (5, -4) \), lub vector \( \vec{AB} \) tsis sib npaug rau lub vector \( \vec{AC} \).
Lo Lus Nug 2: Tsim Cov Vectors Sib Npaug
Txheeb xyuas qhov taw tes \(D\) kom tus vector \(\vec{AB} = \vec{CD}\) nrog qhov taw tes \(C(4, -2)\), qhov taw tes \(B(8, 3)\), thiab \(A(2, 1)\).
Kev Sib Tham:
- Txheeb xyuas tus vector \( \vec{AB} \):
\[
\vec{AB} = (8 - 2, 3 - 1) = (6, 2)
\]
Vim tias \( \vec{CD} \) yuav tsum sib npaug rau \( \vec{AB} \), ces:
\[
\vec{CD} = \vec{AB} = (6, 2)
\]
- Xav tias \( D(x, y) \). Ces \( \vec{CD} = (x – 4, y + 2) \). Los ntawm no peb tau txais:
\[
(x – 4, y + 2) = (6, 2)
\]
Los ntawm kev sib npaug ntawm cov khoom tsim nyog, peb tau txais:
\[
x – 4 = 6 \quad \Rightarrow \quad x = 10
\]
\[
y + 2 = 2 \quad \Rightarrow \quad y = 0
\]
Yog li, qhov taw tes \(D\) yog \((10, 0)\).
Lo lus nug 3: Pov thawj nrog Vector Magnitude
Ua pov thawj tias cov vectors \( \vec{PQ} \) thiab \( \vec{RS} \) yog sib npaug, muab \( P(1, 2) \), \( Q(4, 6) \), \( R(-3, -7) \), thiab \( S(0, -3) \).
Kev Sib Tham:
- Txheeb xyuas tus vector \( \vec{PQ} \):
\[
\vec{PQ} = (4 - 1, 6 - 2) = (3, 4)
\]
- Txhais lub vector \( \vec{RS} \):
\[
\vec{RS} = (0 – (-3), -3 – (-7)) = (3, 4)
\]
Los ntawm cov txiaj ntsig ntawm kev xam, peb pom tias \( \vec{PQ} = (3, 4) \) thiab \( \vec{RS} = (3, 4) \). Vim tias ob qho vectors muaj cov khoom sib xws, \( \vec{PQ} \) yog sib npaug rau \( \vec{RS} \).
Kev Siv Cov Vectors Sib Npaug
Cov vectors sib npaug feem ntau siv rau hauv ntau yam kev kawm txog kev tshawb fawb. Hauv physics, lawv siv los txhais cov zog lossis kev hloov chaw uas muaj tib qhov loj thiab kev coj. Hauv cov duab computer, cov vectors siv los hloov pauv thiab ua kom cov khoom duab zoo.
Xaus
Kev nkag siab txog lub tswv yim ntawm cov vectors sib npaug hauv Cartesian coordinate system yog lub hauv paus tseem ceeb rau kev suav lej thiab nws cov ntawv thov dav dua. Tsab xov xwm no tau tham txog yuav ua li cas txiav txim siab cov vectors sib npaug los ntawm ntau qhov teeb meem piv txwv thiab lawv cov kev daws teeb meem. Los ntawm kev nkag siab thiab siv lub tswv yim no, peb tuaj yeem daws ntau yam teeb meem uas cuam tshuam nrog kev tshuaj xyuas vector hauv ntau qhov chaw ntawm kev tshawb fawb.
Peb vam tias qhov kev sib tham no yuav pab koj nkag siab txog lub tswv yim ntawm cov vectors sib npaug hauv Cartesian coordinate system. Zoo siab rau kev kawm, thiab xav kom koj muaj hmoo hauv kev paub txog cov vectors!
