A cikin ilmin lissafi, gangara ko gradient na layi lamba ce da ke bayyana alkibla da tsayin layin (yana kururuwa Wikipedia). Ana ƙididdige shi ta hanyar gano rabon canji a cikin y-coordinate zuwa canji a cikin haɗin-x tsakanin maɓalli guda biyu akan layi.

Misali, idan kuna da maki biyu akan layi, (1,2) da (3,4), gangaren layin tsakanin su shine (4-2)/(3-1) = 2/2 = 1. Za mu isa ga wannan ba da jimawa ba.

Slope muhimmin ra'ayi ne a cikin ilimin lissafi kuma yana da aikace-aikace na zahiri da yawa. Alal misali, ana iya amfani da shi don ƙididdige saurin abu, yawan canjin aiki, ko kuma tsayin tudu.

A cikin duniyar gaske, ana amfani da gangare a fagage daban-daban kamar ilimin ƙasa, injiniyan farar hula, gine-gine, da kimiyyar lissafi. A cikin labarin ƙasa, ana amfani da gangare don kwatanta gangaren saman ƙasa. Ana amfani da shi don yin samfuri na zubar da ruwa a saman ƙasa, siffanta wurin zama, rarraba ƙasa, tantance yuwuwar haɓakawa, da ƙirar haɗarin gobarar daji.

A aikin injiniyan farar hula, ana amfani da gangare wajen tsara hanyoyi, gadoji, da sauran gine-gine. Ana amfani da shi don ƙayyade hanya mafi kyau don kammala aikin da gina matakan hawan keken hannu, hanyoyi, da matakala.

A cikin gine-gine, ana amfani da gangara don tsara gine-gine da gine-ginen da suke da kwanciyar hankali da aminci. A cikin ilimin kimiyyar lissafi, ana amfani da gangara don bayyana saurin abu a kan lokaci.

Ina nufin magana mai mahimmanci…

Asalin Ka'idodin Tudu

Ana ƙididdige gangara azaman rabon canjin tsaye (tashi) zuwa canjin kwance (gudu) tsakanin maki biyu akan layi.

An bayyana dabarar gangare a matsayin m = (y2 – y1) / (x2 – x1).

A cikin dabarar da ke sama, akwai maki biyu, yanzu kowane batu yana da duka madaidaicin y valve da ƙimar x. Haɗin kai na batu1 shine (x1, y1) kuma na batu2 shine (x2, y2) kamar yadda aka nuna a cikin adadi na sama.

Akwai nau'ikan gangara guda huɗu: tabbatacce, korau, sifili, da waɗanda ba a bayyana su ba.

Kyakkyawan gangara yana nuna cewa layin yana ƙaruwa daga hagu zuwa dama, yayin da mummunan gangara yana nuna cewa layin yana raguwa daga hagu zuwa dama.

Kuskuren sifili yana nuna cewa layin a kwance yake, yayin da gangaren da ba a bayyana ba yana nuna cewa layin yana tsaye.

Hoton da ke ƙasa yana kwatanta nau'ikan gangara daban-daban:

Ƙididdiga Rarraba: Jagorar Mataki-by-Taki

A cikin wannan sashe, za mu shiga cikin jagorar mataki-mataki kan yadda ake lissafin gangara

A ƙasa akwai jagorar mataki-mataki kan yadda ake lissafin gangara:

1. Gano maki biyu akan layi.
2. Zaɓi aya ɗaya don zama (x1, y1) ɗayan kuma don zama (x2, y2).
3. Nemo canji a tsaye (tashi) ta hanyar rage y-coordinates na maki biyu.
4. Nemo canjin kwance (gudu) ta hanyar cire haɗin haɗin-x na maki biyu.
5. Raba canjin tsaye ta hanyar canjin kwance (tashi sama da gudu) don samun gangara.

Ga misali don kwatanta matakan da ke sama:

A ce muna da maki biyu akan layi, (1, 2) da (3, 6).

Za mu iya lissafin gangaren layin kamar haka:

1. Gano maki biyu akan layi: (1, 2) da (3, 6).
2. Zaɓi aya ɗaya don zama (x1, y1) ɗayan kuma don zama (x2, y2): Bari mu zaɓi (1, 2) azaman (x1, y1) da (3, 6) azaman (x2, y2).
3. Nemo canji a tsaye (tashi) ta hanyar rage y-coordinates na maki biyu: 6 - 2 = 4.
4. Nemo canji a kwance (gudu) ta hanyar cire haɗin haɗin-x na maki biyu: 3 - 1 = 2.
5. Raba canjin tsaye ta hanyar canjin kwance (tashi kan gudu) don samun gangara: 4/2 = 2.

Saboda haka, gangaren yana 2. Ie tabbatacce gangara

Ga wani misali don kwatanta matakan da ke sama:

A ce muna da maki biyu akan layi, (3, 7) da (1, 10).

Za mu iya lissafin gangaren layin kamar haka:

1. Gano maki biyu akan layi: (3, 7) da (1, 10).
2. Zaɓi aya ɗaya don zama (x1, y1) ɗayan kuma don zama (x2, y2): Bari mu zaɓi (3, 7) azaman (x1, y1) da (1, 10) azaman (x2, y2).
3. Nemo canji a tsaye (tashi) ta hanyar rage y-coordinates na maki biyu: 10 - 7 = 3.
4. Nemo canji a kwance (gudu) ta hanyar cire haɗin haɗin-x na maki biyu: 1 - 3 = -2.
5. Raba canjin tsaye ta hanyar canjin kwance (tashi kan gudu) don samun gangara: 3 / -2 = -1.5.

Saboda haka, a cikin hunturu - 1.5. Ie korau gangara.

Anan akwai wasu shawarwari don guje wa kuskuren gama gari yayin ƙididdige gangara:

1. Fahimtar manufar gangara: Ana ƙididdige gangara azaman rabon canji a y zuwa canjin x. Madaidaicin gangare yana nuna haɓakar haɓakawa, yayin da mummunan gangara yana nuna yanayin ƙasa.
2. Sau biyu duba lissafin ku: Lissafin gangara na iya zama da wahala, don haka yana da mahimmanci don bincika aikinku sau biyu. Tabbatar cewa kuna da madaidaitan ƙimar canjin y da canjin x, kuma kun raba su daidai.
3. Yi amfani da Kalkuleta mai gangara: Yin amfani da kalkuleta mai gangara zai rage kurakurai sosai.

Ga wani Kalkuleta mai gangara wanda zaku iya amfani dashi don lissafin gangara ko gradient tsakanin maki biyu a cikin tsarin haɗin gwiwar Cartesian. 

Duk abin da za ku yi lokacin amfani da wannan madaidaicin kalkuleta shine shigar da ƙimar x1, x2, y1, y2. 

Kalkuleta zai lissafta gangara ta atomatik, ma'auni na layi, tashi, gudu, nisa tsakanin maki biyu, da ƙari mai yawa, ba dole ba ne ku yi kiftawa sau biyu.

Zurfafa cikin Geometry

Kamar yadda muka fada a baya, gangare shine ma'auni na tsayin layi.

A cikin triangles, ana iya amfani da gangaren layin don ƙididdige kwana tsakanin layin da axis x.

Hakanan za'a iya amfani da gangaren layi don tantance ko layi biyu suna layi ɗaya ko a tsaye. Layuka biyu suna layi daya idan suna da gangaren gangare iri daya, kuma suna tsaye ne idan gangar jikinsu maras kyau ce ta juna.

Aikace-aikace na Duniya na Gaskiya

• Gina da Gine-gine: Ana amfani da lissafin gangara wajen zayyana ramuka, matakala, da rufin rufi. Matsayin rufin, alal misali, yana ƙayyade adadin kayan da za a yi amfani da su don gina rufin da kuma aikin rufin.

• Physics: Ana amfani da lissafin gangara a cikin zane-zane na motsi da karfi. Misali, gangaren jadawali-lokaci yana ba da saurin abu.
• tattalin arziki: Ana amfani da lissafin gangara don fahimtar abubuwan da ke faruwa. Alal misali, gangaren madaidaicin buƙatun yana ba da ƙimar da adadin ya buƙaci canje-canje dangane da farashi.

Misalai masu Mu'amala da Motsa jiki

Wannan sashe yana ba da misalan misalan ma'amala da motsa jiki don taimakawa ƙarfafa fahimtar ku game da lissafin gangara.

Matsala ta 1:

Yi la'akari da maki biyu akan jirgin sama mai daidaitawa: ( A(2, 5)) da ( B (4, 9)). Yi lissafin gangaren layin da ke wucewa ta waɗannan maki ta amfani da dabarar gangara.

Magani:

m = (9-5) / (4-2) = (4)/(2) = 2

Matsala ta 2:

An ba da maki biyu (C (3, 8)) da (D (7, 2)), lissafta gangaren layin da ke wucewa ta waɗannan maki ta amfani da dabarar gangara.

Magani:

m = (2 - 8) / (7 - 3) = (-6) / (4) = -1.5

Yanayin Rayuwa ta Gaskiya

Nano 1: Tsarin Ramp

Ka yi tunanin kai masanin gine-gine ne da ke da alhakin kera hanyar hawan keken hannu don ƙofar gini. Yi amfani da lissafin gangara don tantance mafi kyawun gangara don samun dama yayin da ake bin ƙa'idodin aminci.

Nano 2: Hanyoyin Tattalin Arziki

A matsayin manazarcin kuɗi, bincika saitin maki bayanan tattalin arziki a kan lokaci kuma ƙididdige gangara don gano abubuwan da ke faruwa. Ta yaya wannan bayanin zai kasance mai mahimmanci don yin tsinkaya na gaskiya?

Yanzu, ƙwallon naku ne don harba, Raba mafita ko hanyoyin da kuka yi amfani da lissafin gangara a rayuwarku. Ko yana sake fasalin lambun ku, ko shan gilashin ruwa.

Jin kyauta don ƙaddamar da mafita ko raba abubuwan da kuka samu.

Kammalawa

Mun zo ƙarshen wannan labarin, bari mu sake duba mahimman abubuwan da aka rubuta a cikin wannan labarin

Makullin Maɓalli:

• gangare yana auna tsayin layi kuma yana da mahimmanci a cikin lissafi da aikace-aikace daban-daban na zahiri.
• Tsarin gangara (m = {y2 - y1} / {x2 - x1})
• Nau'ukan Tudu 4 su ne; Tabbatacce, korau, sifili, da gangaren da ba a bayyana ba kuma kowanne yana ba da bayanai na musamman game da halayen layi.
• A cikin duniyar gaske, ana amfani da gangare a fagage daban-daban kamar ilimin ƙasa, injiniyan farar hula, gine-gine, da kimiyyar lissafi.