Imibuzo Eyisibonelo Exoxa Ngama-Vector Ekholomu Nama-Vector Emigqa
Kumathematika, ikakhulukazi i-algebra eqondile, ama-vector angumqondo oyisisekelo ovame ukusetshenziswa ezinhlotsheni ezahlukene zokusebenza, kusukela ekubumbeni i-physics kuya ekubaleni. Ama-vector amakholomu nama-vector omugqa ayizinhlobo ezimbili zokumelwa kwama-vector, ngayinye inezici zayo kanye nokusetshenziswa kwayo. Lesi sihloko sizoxoxa ngezinkinga zesibonelo kanye nezixazululo zazo ezibandakanya ama-vector amakholomu nama-vector omugqa.
Incazelo yeVektha Yekholomu kanye neVektha Yemigqa
Ngaphambi kokuba singene emibuzweni eyisibonelo kanye nengxoxo yayo, ake siqale sibukeze izincazelo eziyisisekelo zamavekhtha ekholomu kanye namavekhtha emigqa.
– Amavekhtha ekholomu angamavekhtha ahlelwe kukholomu, okungukuthi, ubukhulu obuqondile obuqondile. Isibonelo:
\[
\mathbf{v} = \begin{pmatrix}
4 \\
3 \\
2
\end{pmatrix}
\]
– Amavektha emigqa ayivektha ahlelwe ngemigqa, okungukuthi, ngesilinganiso esisodwa esivundlile. Isibonelo:
\[
\mathbf{w} = \begin{pmatrix} 5 & 1 & 7 \end{pmatrix}
\]
Isibonelo 1: Ukwengeza Amavektha Ekholomu
Umbuzo:
Njengoba kunikezwe ama-vector amabili ekholomu alandelayo:
\[
\mathbf{u} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix}
\]
Bala isamba samavekhtha amabili ekholomu.
Isixazululo:
Ukwengezwa kwamavekhtha amabili ekholomu kwenziwa ngokungeza izakhi zawo ezihambisanayo.
\[
\mathbf{u} + \mathbf{v} = \qala{pmatrix}
1 \\
2 \\
3
\end{pmatrix} + \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix} = \begin{pmatrix}
1 + 4 \\
2 + 1 \\
3 + 0
\end{pmatrix} = \begin{pmatrix}
5 \\
3 \\
3
\end{pmatrix}
\]
Ngakho-ke, isamba sika-\(\mathbf{u}\) kanye no-\(\mathbf{v}\) singu-\(\begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}\).
Isibonelo Umbuzo 2: Ukwengeza Amavektha Emigqa
Umbuzo:
Njengoba kunikezwe ama-vector amabili alandelayo:
\[
\mathbf{a} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 & 3 & 5 \end{pmatrix}
\]
Bala isamba samavekhtha amabili emigqa.
Isixazululo:
Ukwengezwa kwamavektha emigqa emibili kwenziwa ngokungeza izakhi ezihambisanayo.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
Ngakho-ke, isamba sika-\(\mathbf{a}\) kanye no-\(\mathbf{b}\) singu-\(\begin{pmatrix} 3 & 7 & 11 \end{pmatrix}\).
Isibonelo 3: Ukuphindaphinda kwe-Scalar ngamaVektha Ekholomu
Umbuzo:
Uma unikezwe ivektha yekholomu \(\mathbf{c}\) kanye ne-scalar \(k\):
\[
\mathbf{c} = \begin{pmatrix}
-3 \\
4 \\
5
\end{pmatrix}, \quad k = 2
\]
Bala umphumela wokuphindaphinda kwe-scalar.
Isixazululo:
Ukuphindaphinda kwe-scalar nge-vector yekholomu kwenziwa ngokuphindaphinda into ngayinye ye-vector nge-scalar.
\[
k\mathbf{c} = 2 \qala{pmatrix}
-3 \\
4 \\
5
\end{pmatrix} = \begin{pmatrix}
2 \izikhathi -3 \\
Izikhathi ezi-2 ezi-4
2 \izikhathi ezingu-5
\end{pmatrix} = \begin{pmatrix}
-6 \\
8 \\
10
\end{pmatrix}
\]
Ngakho-ke, umphumela wokuphindaphinda i-scalar \(2\) nge-vector yekholomu \(\mathbf{c}\) ngu-\(\begin{pmatrix} -6 \\ 8 \\ 10 \end{pmatrix}\).
Isibonelo Umbuzo 4: Ukuphindaphinda kwe-Scalar ngamaVektha Emigqa
Umbuzo:
Kunikezwe ivektha yomugqa \(\mathbf{d}\) kanye ne-scalar \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 & -2 & 1 \end{pmatrix}, \quad m = -3
\]
Bala umphumela wokuphindaphinda kwe-scalar.
Isixazululo:
Ukuphindaphinda kwe-Scalar ngevektha yomugqa kwenziwa ngokuphindaphinda into ngayinye yevektha nge-scalar.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 & -2 & 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 & -3 \times -2 & -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 & 6 & -3 \end{pmatrix}
\]
Ngakho-ke, umphumela wokuphindaphinda i-scalar \(-3\) nge-vector yomugqa \(\mathbf{d}\) ngu-\(\begin{pmatrix} -21 & 6 & -3 \end{pmatrix}\).
Isibonelo 5: Ukuphindaphinda kwe-Matrix \(1 \izikhathi 3\) ngo \(3 \izikhathi 1\) (Ivektha Yomugqa ngeVektha Yekholomu)
Umbuzo:
Kunikezwe i-vector yomugqa \(\mathbf{e}\) kanye nevektha yekholomu \(\mathbf{f}\):
\[
\mathbf{e} = \qala{pmatrix} 2 & -1 & 4 \end{pmatrix}, \quad \mathbf{f} = \qala{pmatrix}
5 \\
3 \\
-2
\end{pmatrix}
\]
Bala umkhiqizo wamavektha amabili.
Isixazululo:
Ukuze kwenziwe ukuphindaphinda kwe-matrix, ivektha yomugqa \(\mathbf{e}\) iphathwa njenge-matrix \(1 \times 3\), kanti ivektha yekholomu \(\mathbf{f}\) iphathwa njenge-matrix \(3 \times 1\). Umphumela walokhu kuphindaphinda yi-scalar, okungukuthi isamba semikhiqizo yezinto ezihambisanayo:
\[
\mathbf{e} \mathbf{f} = \qala{pmatrix} 2 & -1 & 4 \end{pmatrix} \qala{pmatrix}
5 \\
3 \\
-2
\end{pmatrix} = (2 \izikhathi 5) + (-1 \izikhathi 3) + (4 \izikhathi -2) = 10 – 3 – 8 = -1
\]
Ngakho-ke, umphumela wokuphindaphinda ivektha yomugqa \(\mathbf{e}\) ngevektha yekholomu \(\mathbf{f}\) ngu \(-1\).
Isibonelo 6: Ukuphindaphinda kwe-Matrix \(3 \izikhathi 1\) ngo \(1 \izikhathi 3\) (I-Column Vector nge-Row Vector)
Umbuzo:
Uma unikezwe ivekhtha yekholomu \(\mathbf{g}\) kanye nevekhtha yomugqa \(\mathbf{h}\):
\[
\mathbf{g} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}
\]
Bala umkhiqizo wamavektha amabili.
Isixazululo:
Ukuphindaphinda kwe-matrix kwe-column vector nge-row vector kukhiqiza i-(\(3 \times 1\)) matrix ephindaphindwe yi-(\(1 \times 3\)) ekhiqiza i-\(3 \times 3\) matrix. I-element ngayinye entsha ingumkhiqizo we-element zayo ezihambisanayo:
\[
\mathbf{g} \mathbf{h} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix}
1 \izikhathi 4 & 1 \izikhathi 5 & 1 \izikhathi 6 \\
2 \izikhathi 4 & 2 \izikhathi 5 & 2 \izikhathi 6 \\
3 \izikhathi 4 & 3 \izikhathi 5 & 3 \izikhathi 6
\end{pmatrix} = \begin{pmatrix}
4 kanye no-5 kanye no-6 \\
8 kanye no-10 kanye no-12 \\
12 & 15 & 18
\end{pmatrix}
\]
Ngakho-ke, umphumela wokuphindaphinda ivektha yekholomu \(\mathbf{g}\) ngevektha yomugqa \(\mathbf{h}\) yi-matrix:
\[
\begin{pmatrix}
4 kanye no-5 kanye no-6 \\
8 kanye no-10 kanye no-12 \\
12 & 15 & 18
\end{pmatrix}
\]
Isiphetho
Kuyo yonke le ndatshana, sibone izibonelo eziningana ezihilela amavekhtha ekholomu kanye nomugqa. Ukwengezwa kwamavekhtha ekholomu kanye nomugqa kufezwa ngokungeza izakhi zawo ezihambisanayo. Ukuphindaphinda kwe-scalar ngevekhtha nakho kufezwa ngokuphindaphinda isakhi ngasinye sevekhtha nge-scalar. Okokugcina, sifunde ukuthi singaphindaphinda kanjani amavekhtha emigqa kanye nekholomu, siveze i-scalar noma i-matrix, kuye ngokuhleleka kwawo. Ukuqonda kahle le misebenzi eyisisekelo kubalulekile ezinhlelweni zokusebenza eziyinkimbinkimbi kakhulu ku-algebra eqondile kanye nokuhlaziywa kwedatha.
