Misali na Matsalolin Tattaunawa Kan Vectors Masu Juyawa
Vector abu ne na lissafi wanda ke da girma da alkibla. A cikin nazarin vectors, sau da yawa muna haɗuwa da vectors masu takamaiman halaye. Wani muhimmin ra'ayi a cikin vectors shine vector mai juyawa, ko vector mara kyau. Wannan labarin zai ƙunshi misalai da tattaunawa kan vectors mai juyawa.
Fahimtar Vectors Masu Juyawa
Vektor mai juyawa, wanda galibi ake kira vektor mai rashin kyau, vektor ne wanda yake da girma iri ɗaya amma akasin alkiblar vektor na asali. Idan vektor an nuna shi da \(\vec{a}\), to vektor mai juyawa shine \(-\vec{a}\). A lissafi, idan \(\vec{a} = (a_1, a_2, a_3)\), to \(-\vec{a} = (-a_1, -a_2, -a_3)\).
Misali Tambaya ta 1
Idan aka ba da vector \(\vec{a} = (3, 4, -2)\). Ƙayyade vector mai juyi na \(\vec{a}\).
Tattaunawa:
Domin tantance vector mai juyi na \(\vec{a}\), kawai muna buƙatar canza kowanne daga cikin sassan vector ɗinsa zuwa ga waɗanda ba su da kyau:
\[
-\vec{a} = (-3, -4, 2)
\]
Don haka, juzu'in vector na \(\vec{a} = (3, 4, -2)\) shine \(-\vec{a} = (-3, -4, 2)\).
Misali Tambaya ta 2
Bari vector \(\vec{b} = (7, -5, 0)\). Nemo vector mai juyi na \(\vec{b}\) kuma tabbatar da cewa \(\vec{b} + (-\vec{b}) = \vec{0}\).
Tattaunawa:
Da farko, mun bayyana juzu'in vector na \(\vec{b}\):
\[
-\vec{b} = (-7, 5, 0)
\]
Na gaba, za mu tabbatar da cewa ƙarin vector \(\vec{b}\) da vector ɗinsa na juyewa ya haifar da vector sifili:
\[
\vec{b} + (-\vec{b}) = (7, -5, 0) + (-7, 5, 0)
\]
Mun ƙara abubuwan da ke cikin vector:
\[
(7 – 7, -5 + 5, 0 + 0) = (0, 0, 0)
\]
Don haka, \(\vec{b} + (-\vec{b}) = \vec{0}\), an tabbatar da cewa sakamakon ƙara vector \(\vec{b}\) da vector ɗinsa na juyewa shine vector sifili.
Misali Tambaya ta 3
Idan aka ba da vectors \(\vec{u} = (2, -1)\) da \(\vec{v} = (-2, 1)\). Shin \(\vec{u}\) vector mai juyi na \(\vec{v}\) ne?
Tattaunawa:
Domin tantance ko \(\vec{u}\) da \(\vec{v}\) vectors ne masu juyawa, muna buƙatar duba ko \(\vec{v} = -\vec{u}\).
Lissafi \(-\vec{u}\):
\[
-\vec{u} = (-2, 1)
\]
Sai ya zama cewa \(-\vec{u} = \vec{v}\), wannan yana nufin cewa vector \(\vec{u}\) hakika vector ne mai juyewa na vector \(\vec{v}\).
Misali Tambaya ta 4
Idan an san cewa vektor \(\vec{w}\) yana da girman 5 da kuma alkiblar da ke gaban vektor \(\vec{p} = (4, 3)\), a tantance \(\vec{w}\) a cikin siffar sashi.
Tattaunawa:
Da farko, mun sami girman vector \(\vec{p}\):
\[
|\vec{p}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
Tunda \(\vec{w}\) yana da girma iri ɗaya da \(\vec{p}\) amma akasin haka, to:
\[
\vec{w} = -\vec{p} = (-4, -3)
\]
Don haka, vector \(\vec{w}\) a cikin nau'in sashi shine \(\vec{w} = (-4, -3)\).
Misali Tambaya ta 5
An bayar da maki A(2, 3) da maki B(4, 7). Kayyade vector ɗin matsayi daga maki A zuwa maki B da vector ɗin da yake akasin wannan vector.
Tattaunawa:
Matsayin vector daga aya A zuwa aya B:
\[
\vec{AB} = (B_x – A_x, B_y – A_y) = (4 – 2, 7 – 3) = (2, 4)
\]
Juyin juya halin \(\vec{AB}\):
\[
-\vec{AB} = (-2, -4)
\]
Don haka, vektor ɗin da ke juye da vektor ɗin matsayi \(\vec{AB} = (2, 4)\) shine \(-\vec{AB} = (-2, -4)\).
Misali Tambaya ta 6
Idan aka ba da vector \(\vec{m} = (x, y)\) kuma vector mai juyi na \(\vec{m}\) shine \( (-5, 12)\). Kayyade ƙimar x da y.
Tattaunawa:
Juyin juya halin \(\vec{m}\) shine \( (-x, -y) \), kuma bisa ga matsalar, \((-x, -y) = (-5, 12)\).
Ta hanyar daidaita abubuwan da ke cikin vector, mun sami:
\[
-x = -5 \yana nufin x = 5
\]
\[
-y = 12 \yana nufin y = -12
\]
Don haka, ƙimar \(x\) ita ce 5 kuma ƙimar \(y\) ita ce -12.
Kammalawa
Vectors masu juyawa sune vectors masu girma iri ɗaya amma akasin haka. Ta hanyar fahimtar manufar vectors masu juyawa, za mu iya magance matsaloli daban-daban da suka shafi vectors, kamar tantance mummunan vector, tabbatar da ƙara vectors zuwa sifili, da sauransu. Ana sa ran tattaunawa kan misalan matsalolin da ke sama zai ƙara mana ilimi da fahimtar aiki da vectors masu juyawa.
