Eisimpleirean de Cheistean a’ Deasbad Vectaran Colbh is Vectaran Sreath
Ann am matamataig, gu h-àraidh ailseabra loidhneach, tha vectaran nam bun-bheachd a thathas a’ cleachdadh gu tric ann an diofar thagraidhean, bho mhodaladh fiosaig gu àireamhachadh. Tha vectaran colbh agus vectaran sreath nan dà sheòrsa de riochdachadh vectar, gach fear le na feartan agus na cleachdaidhean aige fhèin. Bruidhnidh an t-artaigil seo air eisimpleirean de dhuilgheadasan agus na fuasglaidhean aca a tha a’ toirt a-steach vectaran colbh agus vectaran sreath.
Mìneachadh air Vector Colbh agus Vector Sreath
Mus tèid sinn a-steach do na ceistean eisimpleir agus an deasbad mun deidhinn, leig dhuinn ath-sgrùdadh a dhèanamh an toiseach air na mìneachaidhean bunaiteach air vectaran colbh agus vectaran sreath.
– Is e vectaran colbh vectaran air an cur ann an colbh, is e sin, aon tomhas dìreach. Eisimpleir:
\[
\mathbf{v} = \begin{pmatrix}
4 \\
3 \\
2
\end{pmatrix}
\]
– Is e vectaran sreathan vectaran air an cur ann an sreathan, is e sin, ann an aon tomhas còmhnard. Eisimpleir:
\[
\mathbf{w} = \begin{pmatrix} 5 & 1 & 7 \end{pmatrix}
\]
Eisimpleir 1: A’ cur Vectaran Colbh ris
Ceist:
Air a thoirt seachad leis an dà vectar colbh a leanas:
\[
\mathbf{u} = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix}
\]
Obraich a-mach suim an dà vectar colbh.
Fuasgladh:
Thèid dà vectar colbh a chur ri chèile le bhith a’ cur nan eileamaidean co-fhreagarrach aca ri chèile.
\[
\mathbf{u} + \mathbf{v} = \toiseach{pmatrix}
1 \\
2 \\
3
\end{pmatrix} + \begin{pmatrix}
4 \\
1 \\
0
\end{pmatrix} = \begin{pmatrix}
1 + 4
2 + 1
3 + 0
\end{pmatrix} = \begin{pmatrix}
5 \\
3 \\
3
\end{pmatrix}
\]
Mar sin, is e suim \(\mathbf{u}\) agus \(\mathbf{v}\) \(\begin{pmatrix} 5 \\ 3 \\ 3 \\ end{pmatrix}\).
Eisimpleir Ceist 2: A’ cur Vectaran Sreath ris
Ceist:
Air a thoirt seachad leis an dà shreath vectar a leanas:
\[
\mathbf{a} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 & 3 & 5 \end{pmatrix}
\]
Obraich a-mach suim an dà vectar sreath.
Fuasgladh:
Thèid dà vectar sreath a chur ri chèile le bhith a’ cur nan eileamaidean co-fhreagarrach ri chèile.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
Mar sin, is e suim \(\mathbf{a}\) agus \(\mathbf{b}\) \(\begin{pmatrix} 3 & 7 & 11 \end{pmatrix}\).
Eisimpleir 3: Iomadachadh Sgalar le Vectaran Colbh
Ceist:
Air a thoirt seachad le vectar colbh \(\mathbf{c}\) agus scalar \(k\):
\[
c = pmatrix
-3 \\
4 \\
5
\end{pmatrix}, \quad k = 2
\]
Obraich a-mach toradh an iomadachaidh scalar.
Fuasgladh:
Thèid iomadachadh sgalar le vectar colbh a dhèanamh le bhith a’ iomadachadh gach eileamaid den vectar le sgalar.
\[
k\mathbf{c} = 2 \toiseach{pmatrix}
-3 \\
4 \\
5
\end{pmatrix} = \begin{pmatrix}
2 uair -3
2 uair 4
2 uair 5
\end{pmatrix} = \begin{pmatrix}
-6 \\
8 \\
10
\end{pmatrix}
\]
Mar sin, is e toradh iomadachadh an scalar _(2) leis a’ vectar colbh _(_mathbf{c}) _(_begin{pmatrix} -6 __8 __10 __end{pmatrix})_.
Eisimpleir Ceist 4: Iomadachadh Sgalar le Vectaran Sreath
Ceist:
Le vectar sreath \(\mathbf{d}\) agus scalar \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 & -2 & 1 \end{pmatrix}, \quad m = -3
\]
Obraich a-mach toradh an iomadachaidh scalar.
Fuasgladh:
Thèid iomadachadh sgalar le vectar sreath a dhèanamh le bhith a’ iomadachadh gach eileamaid den vectar le sgalar.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 & -2 & 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 & -3 \times -2 & -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 & 6 & -3 \end{pmatrix}
\]
Mar sin, is e toradh iomadachadh an scalar \(-3\) leis a’ vectar sreath \(\mathbf{d}\) \(\begin{pmatrix} -21 & 6 & -3 \end{pmatrix}\).
Eisimpleir 5: Iomadachadh Maitrís 1 x 3 le 3 x 1 (Vector Sreath air Vector Colbh)
Ceist:
Le vector sreath \(\mathbf{e}\) agus vector colbh \(\mathbf{f}\):
\[
\mathbf{e} = \toiseach{pmatrix} 2 & -1 & 4 \end{pmatrix}, \quad \mathbf{f} = \toiseach{pmatrix}
5 \\
3 \\
-2
\end{pmatrix}
\]
Obraich a-mach toradh an dà vectar.
Fuasgladh:
Gus iomadachadh maitrís a dhèanamh, thèid dèiligeadh ris a’ vectar sreath \(\mathbf{e}\) mar mhaitris \(1 \times 3\), agus thèid dèiligeadh ris a’ vectar colbh \(\mathbf{f}\) mar mhaitris \(3 \times 1\). Is e sgalar toradh an iomadachaidh seo, is e sin suim thoraidhean nan eileamaidean co-fhreagarrach:
\[
\mathbf{e} \mathbf{f} = \toiseach{pmatrix} 2 & -1 & 4 \end{pmatrix} \toiseach{pmatrix}
5 \\
3 \\
-2
\end{pmatrix} = (2 × 5) + (-1 × 3) + (4 × -2) = 10 – 3 – 8 = -1
\]
Mar sin, is e -1 toradh iomadachadh a’ vectar sreath \(\mathbf{e}\) leis a’ vectar colbh \(\mathbf{f}\).
Eisimpleir 6: Iomadachadh Maitrís 3 x 1 le 1 x 3 (Vector Colbh air Vector Sreath)
Ceist:
Air a thoirt seachad vectar colbh \(\mathbf{g}\) agus vectar sreath \(\mathbf{h}\):
\[
g = pmatrix
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}
\]
Obraich a-mach toradh an dà vectar.
Fuasgladh:
Bidh iomadachadh maitrís vectar colbh le vectar sreath a’ toirt a-mach maitrís (\(3 \times 1\)) air iomadachadh le (\(1 \times 3\)) a bheir a-mach maitrís \(3 \times 3\). Is e toradh nan eileamaidean co-fhreagarrach a th’ ann an gach eileamaid ùr:
\[
g h = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix}
1 uair 4 & 1 uair 5 & 1 uair 6
2 uair 4 & 2 uair 5 & 2 uair 6
3 uairean 4 & 3 uairean 5 & 3 uairean 6
\end{pmatrix} = \begin{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12 & 15 & 18
\end{pmatrix}
\]
Mar sin, is e toradh iomadachadh a’ vectar colbh \(\mathbf{g}\) leis a’ vectar sreath \(\mathbf{h}\) am maitrís:
\[
\begin{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12 & 15 & 18
\end{pmatrix}
\]
Co-dhùnadh
Tron artaigil seo, tha sinn air grunn eisimpleirean fhaicinn a’ toirt a-steach vectaran colbh is sreath. Bithear a’ dèanamh cur-ris de vectaran colbh is sreath le bhith a’ cur an eileamaidean co-fhreagarrach ris. Bithear cuideachd a’ dèanamh iomadachadh sgalar le vectar le bhith ag iomadachadh gach eileamaid den vectar leis an sgalar. Mu dheireadh, tha sinn air ionnsachadh mar a nì sinn iomadachadh air vectaran sreath is colbh, a’ toirt a-mach sgalar no maitrís, a rèir an òrdugh. Tha e deatamach na h-obrachaidhean bunaiteach seo a mhaighstireachd airson tagraidhean nas iom-fhillte ann an ailseabra loidhneach agus mion-sgrùdadh dàta.
