Su'aalo Tusaale ah oo Ka Hadlaya Isbeddelka Diyaaradda Cartesian
Isbeddellada ku yimaada diyaaradda Cartesian waa mowduuc muhiim u ah xisaabta, gaar ahaan joomatari. Isbeddelladani waxay ku lug leeyihiin hawlgallo kala duwan sida turjumaadda, milicsiga, wareegga, iyo ballaarinta, kuwaas oo ujeedadoodu tahay in la dhaqaajiyo ama la beddelo qaabka shay ku jira diyaarad laba-geesood ah. Maqaalkani wuxuu dib u eegi doonaa dhowr dhibaato oo tusaale ah wuxuuna ka hadli doonaa iyaga oo la xiriira isbeddellada diyaaradda Cartesian.
Noocyada Isbeddelka
Kahor inta aynaan guda gelin dhibaatada tusaalaha ah, aan marka hore dib u eegno noocyada isbeddelka soo socda:
1. Turjumaad (Shift)
Turjumaadda waa isbeddelka barta ama shay ee diyaarad ku jira iyadoo masaafo gaar ah loo marayo jiho gaar ah. Turjumaadda waxaa lagu qeexi karaa sidan:
\[
(x, y) \arrow midig (x+a, y+b)
\]
halkaas oo \(a\) iyo \(b\) ay yihiin masaafadaha barokaca toosan iyo kuwa toosan.
2. Milicsiga
Milicsiga waa milicsiga dhibic ama shay oo ka gudba dhidib, ha ahaado dhidibka x, dhidibka y, ama sadar kale. Tusaale ahaan, milicsiga dhidibka x:
\[
(x, y) \arrow midig (x, -y)
\]
3. Wareeg (Wareejin)
Wareeggu waa wareegga dhibic ama shay ku wareegsan bar dhexe iyadoo la adeegsanayo xagal gaar ah. Wareeg saacad-ka-leexashada ah oo xagal \(\theta\) ah waxaa lagu tilmaami karaa sidan:
\[
(x, y) \arrow (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
\]
4. Ballaarinta (Miisaanka)
Ballaarinta waa isbeddelka cabbirka shay iyadoo la adeegsanayo qodob cabbir gaar ah. Haddii qodobka miisaanku yahay \(k\), balaadhinta waxaa lagu tilmaami karaa sidan:
\[
(x, y) \arrow midig (kx, ky)
\]
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad: Turjumaad
Su'aal:
Ku samee turjumaad barta \(A(2, 3)\) adigoo 5 cutub u wareejinaya dhanka midig iyo 4 cutub kor.
Dood:
U turjumidda cutubyada \(5\) dhanka midig waxay la macno tahay kordhinta isku-duwidda x ee \(5\). Odhaahda "4 cutub kor" waxay la macno tahay kordhinta isku-duwidda y ee \(4\). Natiijada turjumaaddu waa:
\[
(x, y) \arrow midig (x+5, y+4)
\]
Markaa qodobka \(A(2, 3)\) ka dib turjumaadda wuxuu noqonayaa:
\[
(x+5, y+4) \rightarrow (2+5, 3+4) \rightarrow (7, 7)
\]
Markaa, qodobka \(A(2, 3)\) ka dib turjumaadda waa \(A'(7, 7)\).
Su'aal 2: Milicsiga
Su'aal:
Muuji barta \(B(-4, 7)\) ee ku saabsan dhidibka y.
Dood:
Milicsiga ku saabsan dhidibka y wuxuu isu-duwidda x u beddelaa taban ee isu-duwidda x ee asalka ah, halka isu-duwidda y ay weli tahay isla.
\[
(x, y) \arrow midig (-x, y)
\]
Markaa qodobka \(B(-4, 7)\) ka dib milicsiga wuxuu noqonayaa:
\[
(x, y) \rightarrow (-(-4), 7) \rightarrow (4, 7)
\]
Markaa, dhibicda \(B(-4, 7)\) ka dib marka la milicsado dhidibka y waa \(B'(4, 7)\).
Su'aal 3: Wareeg
Su'aal:
U wareeji barta \(C(1, 2)\) dhanka \(90^\circle\) dhanka ka soo horjeeda saacadda iyadoo bartamaha ku taal asalka \((0, 0)\).
Dood:
Wareeg ah \(90^\circle\) oo ka soo horjeeda saacadda waxaa lagu tilmaami karaa sidan:
\[
(x, y) \arrow midig (-y, x)
\]
Markaa dhibicda \(C(1, 2)\) ka dib wareegga waxay noqonaysaa:
\[
(x, y) \arrow midig (-2, 1)
\]
Markaa, dhibicda \(C(1, 2)\) ka dib wareeg ah \(90^\circle\) oo ka soo horjeeda saacadda waa \(C'(-2, 1)\).
Su'aal 4: Ballaarinta
Su'aal:
Ku balaari barta \(D(3, 4)\) iyadoo la adeegsanayo cabirka \(k = 2\).
Dood:
Ballaarinta leh cabirka miisaanka ee \(2\) macnaheedu waa in labada isku-duwayaal lagu dhufto \(2\).
\[
(x, y) \fallaab midig (2x, 2y)
\]
Markaa barta \(D(3, 4)\) ka dib marka la ballaariyo waxay noqotaa:
\[
(x, y) \rightarrow (2 \ jeer 3, 2 \ jeer 4) \rightarrow (6, 8)
\]
Markaa, dhibicda \(D(3, 4)\) ka dib marka la ballaariyo qodobka miisaanka \(2\) waa \(D'(6, 8)\).
Su'aal 5: Isku-darka Isbeddellada
Su'aal:
Ka fakar dhidibka x ka dibna ku fidi iyadoo la adeegsanayo cabirka miisaanka \(k = 0.5\) barta \(E(8, -6)\).
Dood:
Tallaabada ugu horreysa waa in la milicsado dhidibka x:
\[
(x, y) \arrow midig (x, -y)
\]
\[
(8, -6) \arrow-ka midig (8, 6)
\]
Tallaabada labaad waa in la sameeyo ballaarinta iyadoo la adeegsanayo cabirka miisaanka ee \(0.5\):
\[
(x, y) \fallaab midig (0.5x, 0.5y)
\]
\[
(8, 6) \rightarrow (0.5 \times 8, 0.5 \times 6) \rightarrow (4, 3)
\]
Markaa, dhibicda \(E(8, -6)\) ka dib marka la milicsado dhidibka x iyo ballaarinta qodobka miisaanka \(0.5\) waa \(E'(4, 3)\).
Su'aal 6: Isbeddelka Wareegga iyo Turjumaadda
Su'aal:
U wareeji barta \(F(-3, 4)\) adoo adeegsanaya \(180^\circ\) dhanka ka soo horjeeda, ka dibna natiijada u turjum vektor \((2, -1)\).
Dood:
Tallaabada ugu horreysa waa inaad ku wareejiso \(180^\circle\) dhanka ka soo horjeeda saacadda:
\[
(x, y) \arrow midig (-x, -y)
\]
\[
(-3, 4) \arrow-ka midig (3, -4)
\]
Tallaabada labaad waa in turjumaad lagu sameeyo vektorka \((2, -1)\):
\[
(x, y) \arrow midig (x+2, y-1)
\]
\[
(3, -4) \rightarrow (3+2, -4-1) \rightarrow (5, -5)
\]
Markaa, dhibicda \(F(-3, 4)\) ka dib wareegga \(180^\circ\) iyo turjumaadda vector \((2, -1)\) waa \(F'(5, -5)\).
Gabagabo
Isbeddellada ku yimaada diyaaradda Cartesian waa fikrad muhiim ah oo ku saabsan joomatari, oo ka kooban hawlgallo kala duwan sida turjumaadda, milicsiga, wareegga, iyo ballaarinta. Markaan fahamno sida nooc kasta oo isbeddel ah u shaqeeyo, waxaan si fudud u beddeli karnaa booska ama qaabka shay diyaaradda ku jira. Iyada oo loo marayo tusaalooyinka kor ku xusan, waxaan arki karnaa codsiyada wax ku oolka ah ee isbeddellada kala duwan iyo sida loo isku dari karo si loo gaaro isbeddello aad u adag. Waxaan rajeyneynaa, maqaalkani inuu waxtar u lahaa fahamka isbeddellada diyaaradda Cartesian.
