Misalin Tambayoyin Tattaunawa Kan Ayyukan Vector
Ayyukan vector wani muhimmin ra'ayi ne a fannin lissafi wanda ke bayyana akai-akai a fannoni daban-daban na bincike, kamar kimiyyar lissafi, injiniyanci, da kimiyyar kwamfuta. A cikin wannan labarin, za mu tattauna misalai da dama na ayyukan vector da mafita don samar da fahimta mai zurfi da kuma cikakken bayani. Waɗannan misalan za su ƙunshi ayyukan asali kamar ƙari da rage vector, da kuma ayyukan ci gaba kamar ninka scalar da ninka cross-vector.
1. Ƙarin Vector da Ragewa
Misali Tambaya ta 1
An ba da vectors guda biyu A da B a cikin nau'in sashi:
\[ \mathbf{A} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \]
\[ \mathbf{B} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} \]
Lissafa sakamakon ƙari da ragi na vectors guda biyu.
Tattaunawa
Don ƙarin vector, muna ƙara kowane ɓangaren da ya dace da vector guda biyu.
\[ \mathbf{A} + \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \\ -1 + 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 1 \end{pmatrix} \]
Don rage vector, muna cire kowane bangare mai dacewa na vectors guda biyu.
\[ \mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} – \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 – (-1) \\ 3 – 4 \\ -1 – 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ -3 \end{pmatrix} \]
2. Saurin Girma ta hanyar Vektor
Misali Tambaya ta 2
Idan aka ba da vector C da scalar k:
\[ \mathbf{C} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \]
\[ k = 4 \]
Lissafa samfurin scalar na vector C ta hanyar scalar k.
Tattaunawa
Ana yin ninka sikelin ta hanyar ninka kowanne bangare na vector ta hanyar ninka sikelin.
\[ k \mathbf{C} = 4 \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \cdot 1 \\ 4 \cdot (-2) \\ 4 \cdot 3 \end{pmatrix} = \begin{pmatrix} 4 \\ -8 \\ 12 \end{pmatrix} \]
3. Samfurin Dot
Misali Tambaya ta 3
An ba da vectors guda biyu D da E:
\[ \mathbf{D} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} \]
\[ \mathbf{E} = \fara{pmatrix} 1 \\ 0 \\ -1 \ karshen{pmatrix} \]
Lissafa samfurin digo na vectors guda biyu.
Tattaunawa
Ana samun samfurin digo na vector guda biyu ta hanyar ƙara samfuran abubuwan da suka dace.
\[ \mathbf{D} \cdot \mathbf{E} = 3 \cdot 1 + (-2) \cdot 0 + 4 \cdot (-1) = 3 + 0 – 4 = -1 \]
4. Kayayyaki Masu Juyawa
Misali Tambaya ta 4
An ba da vectors guda biyu F da G:
\[ \mathbf{F} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \]
\[ \mathbf{G} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \]
Lissafa samfurin giciye na vectors guda biyu.
Tattaunawa
Ana samun samfurin giciye na vectors guda biyu a cikin sarari mai girma uku ta hanyar amfani da ma'aunin matrix da waɗannan vectors suka samar. An bayar da samfurin giciye ta hanyar dabarar:
\[ \mathbf{F} \times \mathbf{G} = \fara{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \ karshen {vmatrix} \]
Ana iya ƙididdige wannan ta hanyar da ke ƙasa:
\[
\mathbf{F} \times \mathbf{G} = \mathbf{i} \fara{vmatrix} 3 & 4 \\ -1 & 2 \ karshen{vmatrix} - \mathbf{j} \fara{vmatrix} 2 & 4 \\ 1 & 2 \ karshen{vmatrix} + \mathbegin\k & -1 \ karshen {vmatrix}
\]
Lissafin mai ƙayyade kowane ƙaramin matrix:
\[
= \mathbf{i} (3 \cdot 2 – 4 \cdot -1) – \mathbf{j} (2 \cdot 2 – 4 \cdot 1) + \mathbf{k} (2 \cdot -1 – 3 \cdot 1)
\]
\[
= \mathbf{i} (6 + 4) – \mathbf{j} (4 – 4) + \mathbf{k} (-2 – 3)
\]
\[
= \mathbf{i} (10) - \mathbf{j} (0) + \mathbf{k} (-5)
\]
\[
= \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}
\]
Saboda haka, haɗin gwiwar F da G shine:
\[ \mathbf{F} \times \mathbf{G} = \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix} \]
5. Tantance Kusurwar da ke tsakanin Vektoci Biyu
Misali Tambaya ta 5
An ba da vectors guda biyu H da I:
\[ \mathbf{H} = \begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{I} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} \]
Kayyade kusurwar da ke tsakanin vectors guda biyu.
Tattaunawa
Za a iya samun kusurwar \(\theta\) tsakanin vectors guda biyu ta amfani da alaƙar da ke tsakanin samfurin digo da girman vectors guda biyu:
\[ \mathbf{H} \cdot \mathbf{I} = \| \mathbf{H} \| \| \mathbf{I} \| \kos \ta \]
Da farko, ƙididdige samfurin digo \( \mathbf{H} \cdot \mathbf{I} \):
\[ \mathbf{H} \cdot \mathbf{I} = 6 \cdot 1 + 2 \cdot 4 + 3 \cdot (-2) = 6 + 8 – 6 = 8 \]
Na gaba, ƙididdige girman vectors guda biyu:
\[ \| \mathbf{H} \| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]
\[ \| \mathbf{I} \| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \]
Sannan, maye gurbin waɗannan dabi'u a cikin dabarar kusurwa:
\[ \cos \theta = \ frac{\mathbf{H} \cdot \mathbf{I}}{\| \mathbf{H} \| \| \mathbf{I} \|} = \frac{8}{7\sqrt{21}} \]
\[ \theta = \cos^{-1} \left( \frac{8}{7\sqrt{21}} \right) \]
A sakamakon ƙarshe, za mu iya amfani da kalkuleta don nemo ƙimar kusurwar:
\[ \theta \kimanin 73,4^\circle \]
Kammalawa
Manufar ayyukan vector yana da matuƙar muhimmanci a fannin lissafi da kimiyya. Wannan labarin ya tattauna misalai da dama na matsaloli da mafita, tun daga ƙarawa da rage vector, ninka scalar, samfurin digo, samfurin giciye, da kuma tantance kusurwar da ke tsakanin vectors guda biyu. Ta hanyar yin aiki ta waɗannan misalan, muna fatan inganta fahimtar ku game da ayyukan vector da kuma taimaka muku warware matsalolin da suka shafi vectors a cikin yanayi daban-daban.
