Misali na Tambayoyin Tattaunawa Game da Faɗaɗa Lissafi
Faɗaɗawa, ko kuma faɗaɗawa da ragewa, muhimmin ra'ayi ne a cikin tsarin juyi. Wannan ra'ayi yakan bayyana a cikin yanayi daban-daban, kamar nazarin alamu, daidaitawa, da sauran aikace-aikacen rayuwa ta gaske. Wannan labarin zai tattauna manufar faɗaɗawa a cikin lissafi kuma ya ba da misalai da yawa na matsaloli da mafita.
Menene Faɗaɗawa?
Faɗaɗawa wani canji ne na geometric wanda ya ƙunshi faɗaɗawa ko rage adadi ta wani ma'aunin sikelin. A cikin faɗaɗawa, ma'auni akan siffa ta geometric zai canza ko kusa da tsakiyar wurin faɗaɗawa ta hanyar abu ɗaya. Faɗaɗawa na iya faɗaɗawa (ma'aunin sikelin > 1), ragewa (0 < ma'aunin sikelin < 1), ko ma nuna siffa idan ma'aunin sikelin ya kasance mara kyau. Alamomi a Faɗaɗawa Lokacin yin faɗaɗawa, wasu alamun da ake amfani da su sau da yawa sun haɗa da: - Cibiyar Faɗaɗawa (P): Ma'aunin da aka ƙayyade wanda ake amfani da shi azaman nuni a cikin tsarin faɗaɗawa. - Ma'aunin Sikeli (k): Rabon tsayi wanda ke ƙayyade yawan faɗaɗawa ko raguwa da ke faruwa. Idan \( k > 1 \), za a faɗaɗa abu. Idan \( 0 < k < 1 \), za a rage abu.
A ce an faɗaɗa ma'ana ta A tare da daidaitawa (x, y) zuwa ma'ana ta A' tare da ma'ana ta sikelin k da kuma cibiyar faɗaɗawa a O (0,0), sannan daidaitawar A' sune (kx, ky). Tsarin Faɗaɗawa Don nemo sabbin daidaitawar sakamakon faɗaɗawa na wani batu, dabarar da aka yi amfani da ita ita ce: \[A' = (x', y') = (kx, ky) \] inda (x, y) sune ma'ana ta farko ta ma'ana kuma k shine ma'ana ta sikelin. Tambayoyi da Tattaunawa Misali Tambaya 1 Ma'ana ta A (2, 3) an faɗaɗa ta da ma'ana ta sikelin 2 kuma tsakiyar faɗaɗawa tana a O (0, 0). Ƙayyade ma'ana ta ma'ana ta A bayan faɗaɗawa. Tattaunawa: Ma'ana ta ma'ana ta A sune (2, 3), ma'ana ta sikelin ita ce k = 2, kuma tsakiyar faɗaɗawa ita ce ma'ana ta O (0, 0). Bisa ga dabarar faɗaɗawa: \[ A' = (kx, ky) = (2 \cdot 2, 2 \cdot 3) = (4, 6) \] Don haka, daidaitawar maki A bayan faɗaɗawa sune (4, 6). Misali Tambaya ta 2 Maki B (-1, 4) an faɗaɗa shi da ma'aunin sikelin 0,5 kuma tsakiyar faɗaɗawa yana a O (0, 0). Ƙayyade daidaitawar maki B bayan faɗaɗawa. Magani: Daidaitowar maki B sune (-1, 4), ma'aunin sikelin k = 0,5, kuma tsakiyar faɗaɗawa shine maki O (0, 0). Dangane da dabarar faɗaɗawa: \[ B' = (kx, ky) = (0,5 \cdot -1, 0,5 \cdot 4) = (-0,5, 2) \] Don haka, daidaitattun maki B bayan faɗaɗawa sune (-0,5, 2). Misali Tambaya ta 3 Maki C (3, -2) an faɗaɗa shi da ma'aunin sikelin na -1 kuma tsakiyar faɗaɗawa yana a O (0, 0). Ƙayyade daidaitattun maki C bayan faɗaɗawa. Magani: Daidaitowar maki C sune (3, -2), ma'aunin sikelin k = -1, kuma tsakiyar faɗaɗawa yana a ma'aunin O (0, 0). Bisa ga dabarar faɗaɗawa: \[C' = (kx, ky) = (-1 \cdot 3, -1 \cdot -2) = (-3, 2) \] Don haka, daidaitattun maki C bayan faɗaɗawa sune (-3, 2). Misali Tambaya ta 4 Maki D (2, 5) an faɗaɗa shi da ma'aunin sikelin na 3 kuma tsakiyar faɗaɗawa yana a P (1, 1). Ƙayyade daidaitattun maki D bayan faɗaɗawa. Tattaunawa: Daidaitowar maki D sune (2, 5), ma'aunin sikelin k = 3, kuma tsakiyar fadada shine maki P (1, 1). Da farko, muna canza ma'aunin D zuwa tsarin daidaitawa na tsakiyar fadada P (1, 1): Daidaitowar dangantaka ta D zuwa P sune: \[ D_r = (2 - 1, 5 - 1) = (1, 4) \] Yi fadada tare da ma'aunin sikelin k a ma'aunin D_r: \[ D_r' = (kx, ky) = (3 \cdot 1, 3 \cdot 4) = (3, 12) \] A ƙarshe, mun mayar da maki D_r' zuwa tsarin daidaitawa na farko: \[ D' = (D_r' + P) = (3 + 1, 12 + 1) = (4, 13) \] Don haka, daidaitattun maki D bayan faɗaɗa su ne (4, 13). Misali Tambaya ta 5 Maki E (-2, -3) an faɗaɗa shi da ma'aunin sikelin 0,25 kuma tsakiyar faɗaɗa shi ne a P (-1, -1). Ƙayyade daidaitattun maki E bayan faɗaɗa. Magani: Daidaitattun maki E sune (-2, -3), ma'aunin sikelin k = 0,25, kuma tsakiyar faɗaɗa shine ma'aunin P (-1, -1). Da farko, muna canza wurin E zuwa tsarin daidaitawar cibiyar faɗaɗawa P (-1, -1): Daidaito tsakanin E zuwa P sune: \[ E_r = (-2 - (-1), -3 - (-1)) = (-2 + 1, -3 + 1) = (-1, -2) \] Yi faɗaɗawa tare da ma'aunin ma'auni k akan wurin E_r: \[ E_r' = (kx, ky) = (0,25 \cdot -1, 0,25 \cdot -2) = (-0,25, -0,5) \] A ƙarshe, muna mayar da wurin E_r' zuwa tsarin daidaitawa na asali: \[ E' = (E_r' + P) = (-0,25 - 1, -0,5 - 1) = (-1,25, -1,5) \] Don haka, daidaitattun wurin E bayan faɗaɗawa sune (-1,25, -1,5). Kammalawa: Faɗaɗawa wani canji ne na lissafi wanda ke faɗaɗa ko rage siffar lissafi ta wani takamaiman ma'auni. Fahimtar manufar da amfani da faɗaɗawa yana da matuƙar amfani a fannoni daban-daban na karatu, musamman a fannin lissafi. Tare da misalai da tattaunawa, ana fatan wannan zai taimaka wa ɗalibai da masu karatu su fahimci da kuma amfani da manufar faɗaɗawa a cikin yanayi daban-daban na lissafi.