Imibuzo Eyisibonelo Exoxa Ngokuguqulwa Kwendiza YeCartesian
Ukuguqulwa kwendiza yeCartesian kuyisihloko esibalulekile kwizibalo, ikakhulukazi i-geometry. Lokhu kuguqulwa kuhilela imisebenzi ehlukahlukene njengokuhumusha, ukuzindla, ukujikeleza, kanye nokwanda, okuhlose ukuhambisa noma ukushintsha isimo sento endizeni enezinhlangothi ezimbili. Lesi sihloko sizobuyekeza izibonelo eziningana zezinkinga futhi sixoxe ngazo ezihlobene nokuguqulwa kwendiza yeCartesian.
Izinhlobo Zokuguqulwa
Ngaphambi kokuthi singene enkingeni yesibonelo, ake siqale sibukeze izinhlobo ezilandelayo zokuguqulwa:
1. Ukuhumusha (Shift)
Ukuhumusha kungukushintsha kwephuzu noma into endizeni ngebanga elithile endaweni ethile. Ukuhumusha kungachazwa ngokuthi:
\[
(x, y) \umcibisholo ongakwesokudla (x+a, y+b)
\]
lapho \(a\) kanye \(b\) kuyibanga lokuhamba elivundlile neliqondile.
2. Ukuzindla
Ukuzindla kuwukubonakaliswa kwephuzu noma into ngaphesheya kwe-axis, kungaba yi-x-axis, i-y-axis, noma omunye umugqa. Isibonelo, ukuzindla ngaphesheya kwe-x-axis:
\[
(x, y) \umcibisholo ongakwesokudla (x, -y)
\]
3. Ukujikeleza (Spin)
Ukujikeleza kungukujikeleza kwephuzu noma into ezungeze iphuzu eliphakathi nge-engeli ethile. Ukujikeleza okuphikisana newashi nge-engeli \(\theta\) kungachazwa kanje:
\[
(x, y) \umcibisholo ongakwesokudla (x \cos \theta – y \sin \theta, x \sin \theta + y \cos \theta)
\]
4. Ukwanda (Ukwenyuka)
Ukwanda kungukushintsha kosayizi wento ngesici esithile sesikali. Uma isici sesikali singu-\(k\), ukwanda kungachazwa kanje:
\[
(x, y) \umcibisholo ongakwesokudla (kx, ky)
\]
Imibuzo Eyisibonelo Nengxoxo
Umbuzo 1: Ukuhumusha
Umbuzo:
Yenza ukuhumusha endaweni ethi \(A(2, 3)\) ngokushintsha amayunithi ama-5 uye kwesokudla kanye namayunithi ama-4 phezulu.
Ingxoxo:
Ukuhumusha amayunithi angu-\(5\) ngakwesokudla kusho ukukhulisa i-x-coordinate ngo-\(5\). Isitatimende esithi “amayunithi angu-4 phezulu” sisho ukukhulisa i-y-coordinate ngo-\(4\). Umphumela wokuhumusha uthi:
\[
(x, y) \umcibisholo ongakwesokudla (x+5, y+4)
\]
Ngakho iphuzu \(A(2, 3)\) ngemva kokuhumusha liba:
\[
(x+5, y+4) \umcibisholo ongakwesokudla (2+5, 3+4) \umcibisholo ongakwesokudla (7, 7)
\]
Ngakho-ke, iphuzu \(A(2, 3)\) ngemva kokuhumusha lingu \(A'(7, 7)\).
Umbuzo 2: Ukuzindla
Umbuzo:
Cabanga ngephuzu \(B(-4, 7)\) mayelana ne-y-axis.
Ingxoxo:
Ukuzindla nge-y-axis kushintsha i-x-coordinate ibe yi-negative ye-x-coordinate yokuqala, kuyilapho i-y-coordinate ihlala ifana.
\[
(x, y) \umcibisholo ongakwesokudla (-x, y)
\]
Ngakho iphuzu \(B(-4, 7)\) ngemva kokuzindla liba:
\[
(x, y) \umcibisholo ongakwesokudla (-(-4), 7) \umcibisholo ongakwesokudla (4, 7)
\]
Ngakho-ke, iphuzu \(B(-4, 7)\) ngemva kokuzindla nge-y-axis ngu-\(B'(4, 7)\).
Umbuzo 3: Ukujikeleza
Umbuzo:
Jikelezisa iphuzu \(C(1, 2)\) ngo \(90^\circ\) ngokuphambene newashi phakathi nendawo ekuqaleni \((0, 0)\).
Ingxoxo:
Ukujikeleza kwe-\(90^\circ\) ngokuphambene newashi kungachazwa kanje:
\[
(x, y) \umcibisholo ongakwesokudla (-y, x)
\]
Ngakho iphuzu \(C(1, 2)\) ngemva kokujikeleza liba:
\[
(x, y) \umcibisholo ongakwesokudla (-2, 1)
\]
Ngakho-ke, iphuzu \(C(1, 2)\) ngemva kokujikeleza kwe-\(90^\circ\) ngokuphambene newashi lingu-\(C'(-2, 1)\).
Umbuzo 4: Ukwanda
Umbuzo:
Nweba iphuzu \(D(3, 4)\) ngesici sesikali \(k = 2\).
Ingxoxo:
Ukwanda ngesilinganiso se- \(2\) kusho ukuphindaphinda ama-coordinates womabili ngo- \(2\).
\[
(x, y) \umcibisholo ongakwesokudla (2x, 2y)
\]
Ngakho iphuzu \(D(3, 4)\) ngemva kokwandiswa liba:
\[
(x, y) \umcibisholo ongakwesokudla (2 \izikhathi 3, 2 \izikhathi 4) \umcibisholo ongakwesokudla (6, 8)
\]
Ngakho-ke, iphuzu \(D(3, 4)\) ngemva kokwandiswa nge-scale factor \(2\) lingu-\(D'(6, 8)\).
Umbuzo 5: Inhlanganisela Yezinguquko
Umbuzo:
Cabanga nge-x-axis bese unwebeka nge-scale factor \(k = 0.5\) endaweni \(E(8, -6)\).
Ingxoxo:
Isinyathelo sokuqala ukucabanga nge-x-axis:
\[
(x, y) \umcibisholo ongakwesokudla (x, -y)
\]
\[
(8, -6) \umcibisholo ongakwesokudla (8, 6)
\]
Isinyathelo sesibili ukwenza ukunwetshwa ngesilinganiso se- \(0.5\):
\[
(x, y) \umcibisholo ongakwesokudla (0.5x, 0.5y)
\]
\[
(8, 6) \umcibisholo oqondile (0.5 \izikhathi 8, 0.5 \izikhathi 6) \umcibisholo oqondile (4, 3)
\]
Ngakho-ke, iphuzu \(E(8, -6)\) ngemva kokuzindla mayelana ne-x-axis kanye nokwanda nge-scale factor \(0.5\) lingu-\(E'(4, 3)\).
Umbuzo 6: Ukuguqulwa Ngokuzungeza Nokuhumusha
Umbuzo:
Jikelezisa iphuzu \(F(-3, 4)\) ngo \(180^\circ\) ngokuphambene newashi, bese uhumusha umphumela ngevektha \((2, -1)\).
Ingxoxo:
Isinyathelo sokuqala ukuzungeza ngo-\(180^\circ\) ngokuphambene newashi:
\[
(x, y) \umcibisholo ongakwesokudla (-x, -y)
\]
\[
(-3, 4) \umcibisholo ongakwesokudla (3, -4)
\]
Isinyathelo sesibili ukwenza ukuhumusha ngevektha \((2, -1)\):
\[
(x, y) \umcibisholo ongakwesokudla (x+2, y-1)
\]
\[
(3, -4) \umcibisholo ongakwesokudla (3+2, -4-1) \umcibisholo ongakwesokudla (5, -5)
\]
Ngakho-ke, iphuzu \(F(-3, 4)\) ngemva kokujikeleza \(180^\circ\) kanye nokuhumusha nge-vector \((2, -1)\) lingu- \(F'(5, -5)\).
Isiphetho
Ukuguqulwa kwendiza yeCartesian kuwumqondo obalulekile ku-geometry, ohlanganisa imisebenzi ehlukahlukene njengokuhumusha, ukuzindla, ukujikeleza, kanye nokwanda. Ngokuqonda ukuthi uhlobo ngalunye lokuguqulwa lusebenza kanjani, singashintsha kalula isikhundla noma ukuma kwento endizeni. Ngezibonelo ezingenhla, singabona ukusetshenziswa okusebenzayo kokuguqulwa okuhlukahlukene nokuthi kungahlanganiswa kanjani ukuze kufezwe ukuguqulwa okuyinkimbinkimbi kakhulu. Ngethemba ukuthi lesi sihloko sibe usizo ekuqondeni ukuguqulwa kwendiza yeCartesian.
