Ajụjụ Ihe Nlereanya Na-ekwu Maka Mgbanwe n'Ọgbọ Cartesian
Mgbanwe na mbara igwe Cartesian bụ isiokwu dị mkpa na mgbakọ na mwepụ, ọkachasị geometry. Mgbanwe ndị a gụnyere ọrụ dị iche iche dịka ntụgharị asụsụ, ntụgharị uche, ntụgharị uche, na mgbasa, nke ebumnuche ya bụ ịkwaga ma ọ bụ ịgbanwe ọdịdị nke ihe n'ime mbara igwe nwere akụkụ abụọ. Isiokwu a ga-enyocha ọtụtụ nsogbu atụ ma tụlee ha metụtara mgbanwe na mbara igwe Cartesian.
Ụdị Mgbanwe
Tupu anyị abanye n'ihe atụ nsogbu ahụ, ka anyị buru ụzọ lelee ụdị mgbanwe ndị a:
1. Ntụgharị asụsụ (Mgbanwe)
Nsụgharị bụ mgbanwe nke isi ihe ma ọ bụ ihe dị n'ime ụgbọelu site na anya ụfọdụ n'otu ntụziaka. Enwere ike ịkọwa ntụgharị dị ka:
\[
(x, y) \akara aka nri (x+a, y+b)
\]
ebe \(a\) na \(b\) bụ anya nke mgbanwe kwụ ọtọ na nke kwụ ọtọ.
2. Ntụgharị uche
Ntụgharị uche bụ ntụgharị uche nke isi ihe ma ọ bụ ihe gafere axis, ma ọ bụ axis x, axis y, ma ọ bụ ahịrị ọzọ. Dịka ọmụmaatụ, ntụgharị uche n'ofe axis x:
\[
(x, y) \akara aka nri (x, -y)
\]
3. Mgbanwe (Mgbagharị)
Mgbanwe bụ ntụgharị nke isi ihe ma ọ bụ ihe gbara isi ihe dị n'etiti gburugburu site n'akụkụ ụfọdụ. Enwere ike ịkọwa ntụgharị nke akụkụ \(\theta\) n'akụkụ aka ekpe dịka:
\[
(x, y) arrow aka nri (x \cos \ theta – y \ sin \theta, x \ sin \ theta + y \cos \ theta)
\]
4. Mgbasa (Ịhazi)
Mgbasa bụ mgbanwe n'ogo ihe site na ihe nhazi ụfọdụ. Ọ bụrụ na ihe nhazi bụ \(k\), enwere ike ịkọwa mgbasa dị ka:
\[
(x, y) \akara aka nri (kx, ky)
\]
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ajụjụ nke 1: Ntụgharị asụsụ
Ajụjụ:
Mee ntụgharị okwu n'ebe \(A(2, 3)\) site n'ịgbanwe nkeji ise gaa n'aka nri na nkeji anọ elu.
Azịza:
Ịsụgharị nkeji \(5\) gaa n'aka nri pụtara ịbawanye nhazi x site na \(5\). Okwu ahụ bụ "4 nkeji elu" pụtara ịbawanye nhazi y site na \(4\). Nsonaazụ nke ntụgharị ahụ bụ:
\[
(x, y) \akara aka nri (x+5, y+4)
\]
Ya mere, isi ihe \(A(2, 3)\) mgbe ntụgharị asụsụ gasịrị ghọrọ:
\[
(x+5, y+4) \rightarrow (2+5, 3+4) \rightarrow (7, 7)
\]
Ya mere, isi ihe dị na \(A(2, 3)\) mgbe ntụgharị asụsụ gasịrị bụ \(A'(7, 7)\).
Ajụjụ nke Abụọ: Ntụgharị Uche
Ajụjụ:
Cheta isi ihe dị n'etiti \(B(-4, 7)\) gbasara y-axis.
Azịza:
Ntụgharị uche gbasara y-axis na-agbanwe nhazi x ka ọ bụrụ ihe na-adịghị mma nke nhazi x mbụ, ebe nhazi y ka dị otu.
\[
(x, y) \akara aka nri (-x, y)
\]
Ya mere, isi ihe \(B(-4, 7)\) mgbe ntụgharị uche gasịrị na-aghọ:
\[
(x, y) \akara aka nri (-(-4), 7) \akara aka nri (4, 7)
\]
Ya mere, isi ihe \(B(-4, 7)\) mgbe a tụlechara anya gbasara y-axis bụ \(B'(4, 7)\).
Ajụjụ nke 3: Mgbanwe
Ajụjụ:
Tụgharịa isi ihe \(C(1, 2)\) site na \(90^\circ\) n'akụkụ aka ekpe na etiti dị na mmalite \((0, 0)\).
Azịza:
Enwere ike igosi ntụgharị nke \(90^\circ\) n'akụkụ aka ekpe dị ka:
\[
(x, y) \akara aka nri (-y, x)
\]
Ya mere, isi ihe \(C(1, 2)\) mgbe ntụgharị gasịrị na-aghọ:
\[
(x, y) \akara aka nri (-2, 1)
\]
Ya mere, isi ihe \(C(1, 2)\) mgbe agbagharịchara \(90^\circ\) n'akụkụ aka ekpe bụ \(C'(-2, 1)\).
Ajụjụ nke 4: Ịgbasapụ
Ajụjụ:
Mee ka isi ihe \(D(3, 4)\) gbasaa site na iji ihe nha \(k = 2\).
Azịza:
Ịgbasa site na ihe nha nke \(2\) pụtara ịba ụba nhazi abụọ ahụ site na \(2\).
\[
(x, y) \akara aka nri (2x, 2y)
\]
Ya mere, isi ihe \(D(3, 4)\) mgbe e mechachara gbasaa ya ghọrọ:
\[
(x, y) \rightarrow (2 \ugboro 3, 2 \ugboro 4) \rightarrow (6, 8)
\]
Ya mere, isi ihe \(D(3, 4)\) mgbe e mechara gbasaa ya na ihe nha nha \(2\) bụ \(D'(6, 8)\).
Ajụjụ nke Ise: Njikọta nke Mgbanwe
Ajụjụ:
Tụgharịa uche gbasara x-axis wee gbasaa site na iji ihe nha nha \(k = 0.5\) na isi \(E(8, -6)\).
Azịza:
Nzọụkwụ mbụ bụ ịtụgharị uche gbasara x-axis:
\[
(x, y) \akara aka nri (x, -y)
\]
\[
(8, -6) \akara aka nri (8, 6)
\]
Nzọụkwụ nke abụọ bụ ime mgbasa ozi site na iji ihe nha nha nke \(0.5\):
\[
(x, y) \akara aka nri (0.5x, 0.5y)
\]
\[
(8, 6) \rightarrow (0.5 \ugboro 8, 0.5 \ugboro 6) \rightarrow (4, 3)
\]
Ya mere, isi ihe \(E(8, -6)\) mgbe a tụlechara anya gbasara x-axis na mgbasa site na ihe nha nha \(0.5\) bụ \(E'(4, 3)\).
Ajụjụ nke 6: Mgbanwe site na ntụgharị na ntụgharị asụsụ
Ajụjụ:
Tụgharịa isi ihe \(F(-3, 4)\) site na \(180^\circ\) n'akụkụ aka ekpe, wee tụgharịa nsonaazụ ahụ site na vektọ \((2, -1)\).
Azịza:
Nzọụkwụ mbụ bụ ịtụgharị site na \(180^\circ\) megide elekere:
\[
(x, y) \akara aka nri (-x, -y)
\]
\[
(-3, 4) \akara aka nri (3, -4)
\]
Nzọụkwụ nke abụọ bụ ịtụgharị asụsụ site na iji vektọ \((2, -1)\):
\[
(x, y) \akara aka nri (x+2, y-1)
\]
\[
(3, -4) \akara aka nri (3+2, -4-1) \akara aka nri (5, -5)
\]
Ya mere, isi ihe \(F(-3, 4)\) mgbe ntụgharị \(180^\circ\) gasịrị na ntụgharị site na vektọ \((2, -1)\) bụ \(F'(5, -5)\).
Mmechi
Mgbanwe na mbara igwe Cartesian bụ echiche dị mkpa na geometry, gụnyere ọrụ dị iche iche dịka ntụgharị asụsụ, ntụgharị uche, ntụgharị uche, na mgbasa. Site n'ịghọta otu ụdị mgbanwe ọ bụla si arụ ọrụ, anyị nwere ike ịgbanwe ọnọdụ ma ọ bụ ọdịdị nke ihe dị na mbara igwe ngwa ngwa. Site na ihe atụ ndị dị n'elu, anyị nwere ike ịhụ ojiji bara uru nke mgbanwe dị iche iche na otu esi ejikọta ha iji nweta mgbanwe ndị ka mgbagwoju anya. Olileanya, isiokwu a enyerela aka n'ịghọta mgbanwe na mbara igwe Cartesian.
