Tambayoyi Misali Game da Vectors na Ginshiƙi da Vectors na Layi
A fannin lissafi, musamman algebra mai layi, vectors wani muhimmin ra'ayi ne da ake yawan amfani da shi a aikace-aikace daban-daban, tun daga ƙirar kimiyyar lissafi zuwa lissafi. Vectors na ginshiƙi da vectors na layi nau'i biyu ne na wakilcin vector, kowannensu yana da halaye da amfaninsa. Wannan labarin zai tattauna misalai na matsaloli da mafitarsu da suka shafi vectors na ginshiƙi da vectors na layi.
Ma'anar Vektor na Ginshiƙi da Vektor na Layi
Kafin mu shiga cikin tambayoyin misalai da tattaunawarsu, bari mu fara duba ma'anonin asali na vectors na ginshiƙai da vectors na layi.
– Vektocin ginshiƙai vektoci ne da aka shirya a cikin ginshiƙi, wato, girma ɗaya a tsaye. Misali:
\[
\mathbf{v} = \begin{pmatrix}
4 \\
3 \\
2
\end{pmatrix}
\]
– Vektocin layi vektoci ne da aka shirya a layuka, wato, a cikin girma ɗaya a kwance. Misali:
\[
\mathbf{w} = \begin{pmatrix} 5 & 1 & 7 \end{pmatrix}
\]
Misali na 1: Ƙara Vektoran Ginshiƙi
Tambaya:
Ganin waɗannan vectors guda biyu masu zuwa:
\[
\mathbf{u} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix}
\]
Lissafa jimlar vectors guda biyu na ginshiƙai.
Mafita:
Ana yin ƙarin vector guda biyu na ginshiƙai ta hanyar ƙara abubuwan da suka dace.
\[
\mathbf{u} + \mathbf{v} = \fara{pmatrix}
1 \\
2 \\
3
\end{pmatrix} + \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix} = \begin{pmatrix}
1 + 4 \\
2 + 1 \\
3 + 0
\end{pmatrix} = \begin{pmatrix}
5 \\
3 \\
3
\end{pmatrix}
\]
Don haka, jimlar \(\mathbf{u}\) da \(\mathbf{v}\) shine \(\begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}\).
Misali Tambaya ta 2: Ƙara Vectors na Layi
Tambaya:
Ganin waɗannan vectors guda biyu masu zuwa:
\[
\mathbf{a} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 & 3 & 5 \end{pmatrix}
\]
Lissafa jimlar vectors ɗin layuka biyu.
Mafita:
Ana yin ƙarin vectors guda biyu ta hanyar ƙara abubuwan da suka dace.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
Don haka, jimlar \(\mathbf{a}\) da \(\mathbf{b}\) shine \(\begin{pmatrix} 3 & 7 & 11 \end{pmatrix}\).
Misali na 3: Rubutu Mai Sauƙi ta hanyar Vectors na Shafi
Tambaya:
An ba da vector na ginshiƙi \(\mathbf{c}\) da scalar \(k\):
\[
\mathbf{c} = \begin{pmatrix}
-3\\
4 \\
5
\end{pmatrix}, \quad k = 2
\]
Lissafa sakamakon ninka sikelin.
Mafita:
Ana yin ninka sikelin ta hanyar amfani da vector mai shafi ta hanyar ninka kowanne abu na vector da silar.
\[
k\mathbf{c} = 2 \fara{pmatrix}
-3\\
4 \\
5
\end{pmatrix} = \begin{pmatrix}
Sau 2 -3
Sau 2 4
Sau 2 sau 5
\end{pmatrix} = \begin{pmatrix}
-6\\
8 \\
10
\end{pmatrix}
\]
Don haka, sakamakon ninka scalar \(2\) ta hanyar vector na shafi \(\mathbf{c}\) shine \(\begin{pmatrix} -6 \\ 8 \\ 10 \end{pmatrix}\).
Misali Tambaya ta 4: Rubutu Mai Sauƙi ta hanyar Jere Vectors
Tambaya:
An ba da layin vector \(\mathbf{d}\) da scalar \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 & -2 & 1 \end{pmatrix}, \quad m = -3
\]
Lissafa sakamakon ninka sikelin.
Mafita:
Ana yin ninka sikelin ta hanyar amfani da layin vector ta hanyar ninka kowanne abu na vector da sikala.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 & -2 & 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 & -3 \times -2 & -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 & 6 & -3 \end{pmatrix}
\]
Don haka, sakamakon ninka scalar \(-3\) ta hanyar layin vector \(\mathbf{d}\) shine \(\begin{pmatrix} -21 & 6 & -3 \end{pmatrix}\).
Misali na 5: Rufe Matrix \(1 \sau 3\) ta \(3 \sau 1\) (Jerin Vector ta Column Vector)
Tambaya:
An ba da vector jere \(\mathbf{e}\) da ginshiƙi vector \(\mathbf{f}\):
\[
\mathbf{e} = \fara{pmatrix} 2 & -1 & 4 \ karshen{pmatrix}, \quad \mathbf{f} = \fara{pmatrix}
5 \\
3 \\
-2
\end{pmatrix}
\]
Lissafa samfurin vectors guda biyu.
Mafita:
Domin yin ninka matrix, ana ɗaukar vector na layi \(\mathbf{e}\) a matsayin matrix \(1 \times 3\), kuma vector na shafi \(\mathbf{f}\) ana ɗaukarsa a matsayin matrix \(3 \times 1\). Sakamakon wannan ninkawa shine sikelin, wato jimlar samfuran abubuwan da suka dace:
\[
\mathbf{e} \mathbf{f} = \fara{pmatrix} 2 & -1 & 4 \ karshen{pmatrix} \fara{pmatrix}
5 \\
3 \\
-2
\end{pmatrix} = (2 \sau 5) + (-1 \sau 3) + (4 \sau -2) = 10 – 3 – 8 = -1
\]
Don haka, sakamakon ninka layin vector \(\mathbf{e}\) ta hanyar ginshiƙi vector \(\mathbf{f}\) shine \(-1\).
Misali na 6: Rubutuwar Matrix \(3 \sau 1\) ta \(1 \sau 3\) (Vector na shafi ta hanyar Vector mai layi)
Tambaya:
An ba da vector mai shafi mai lamba \(\mathbf{g}\) da vector mai layi mai lamba \(\mathbf{h}\):
\[
\mathbf{g} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}
\]
Lissafa samfurin vectors guda biyu.
Mafita:
Haɓaka matrix na vector na ginshiƙi ta hanyar vector na layi yana samar da matrix (\(3 \sau 1\)) wanda aka ninka ta (\(1 \sau 3\)) wanda ke samar da matrix mai lamba 3 (3 \sau 3 \). Kowane sabon abu samfurin abubuwan da suka dace ne:
\[
\mathbf{g} \mathbf{h} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix}
Sau 1 sau 4 da 1 sau 5 da 1 sau 6 \\
Sau 2 sau 4 da 2 sau 5 da 2 sau 6 \\
Sau 3 sau 4 da 3 sau 5 da 3 sau 6
\end{pmatrix} = \begin{pmatrix}
4 da 5 da 6 \\
8 da 10 da 12 \\
12&15&18
\end{pmatrix}
\]
Don haka, sakamakon ninka vector na ginshiƙi \(\mathbf{g}\) ta hanyar vector na layi \(\mathbf{h}\) shine matrix:
\[
\begin{pmatrix}
4 da 5 da 6 \\
8 da 10 da 12 \\
12&15&18
\end{pmatrix}
\]
Kammalawa
A cikin wannan labarin, mun ga misalai da dama da suka shafi vectors na ginshiƙi da na layi. Ana samun ƙarin vectors na ginshiƙi da na layi ta hanyar ƙara abubuwan da suka dace. Ana kuma samun ninkawa na sikala ta hanyar ninkawa kowane abu na vector ta hanyar sikala. A ƙarshe, mun koyi yadda ake ninka vectors na layi da na ginshiƙi, suna samar da sikala ko matrix, dangane da tsarinsu. Kwarewa a waɗannan ayyukan asali yana da mahimmanci ga aikace-aikace masu rikitarwa a cikin algebra mai layi da nazarin bayanai.
