Su'aalo Tusaale ah oo Ka Hadlaya Vektors-ka Tiirarka iyo Vektors-ka Safka ah
Xisaabta, gaar ahaan aljabrada toosan, vektarrada waa fikrad aasaasi ah oo inta badan loo isticmaalo codsiyada kala duwan, laga bilaabo qaabaynta fiisikiska ilaa xisaabinta. Vektarrada tiirarka iyo vektarrada safka waa laba nooc oo matalaadda vektarrada ah, mid walbana wuxuu leeyahay astaamo iyo isticmaal u gaar ah. Maqaalkani wuxuu ka hadli doonaa masalooyinka tusaalaha ah iyo xalalkooda ku lug leh vektarrada tiirarka iyo vektarrada safka.
Qeexitaanka Vektor-ka Tiirka iyo Vektor-ka Safka
Kahor inta aynaan guda gelin su'aalaha tusaalaha ah iyo dooddooda, aan marka hore dib u eegno qeexitaannada aasaasiga ah ee vectors-ka tiirarka iyo vectors-ka safka.
– Vektorrada tiirarka waa vektorro ku habaysan tiir, taas oo ah, hal cabbir oo toosan. Tusaale:
\[
\mathbf{v} = \bilaw{pmatrix}
4 \\
3 \\
2
\dhammaad{pmatrix}
\]
– Vektorrada safka waa vektorro loo habeeyey saf, taas oo ah, hal cabbir oo siman. Tusaale:
\[
\mathbf{w} = \begin{pmatrix} 5 & 1 & 7 \dhammaad{pmatrix}
\]
Tusaale 1: Ku darista Vektorrada Tiirarka
Su'aal:
Marka la eego labada fallaadhaha tiir ee soo socda:
\[
\mathbf{u} = \bilaw{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix}
4 \\
1 \\
0
\dhammaad{pmatrix}
\]
Xisaabi wadarta labada vector ee tiirarka.
Xalka:
Ku darista laba vector oo tiir ah waxaa lagu sameeyaa iyadoo lagu darayo walxaha u dhigma.
\[
\mathbf{u} + \mathbf{v} = \bilow{pmatrix}
1 \\
2 \\
3
\end{pmatrix} + \begin{pmatrix}
4 \\
1 \\
0
\dhammaad{pmatrix} = \bilaw{pmatrix}
1 + 4 \\
2 + 1 \\
3 + 0
\dhammaad{pmatrix} = \bilaw{pmatrix}
5 \\
3 \\
3
\dhammaad{pmatrix}
\]
Markaa, wadarta guud ee \(\mathbf{u}\) iyo \(\mathbf{v}\) waa \(\begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}\).
Su'aal Tusaale 2: Ku darista Vektorrada Safka
Su'aal:
Marka la eego vector-yada laba saf ee soo socda:
\[
\mathbf{a} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 & 3 & 5 \end{pmatrix}
\]
Xisaabi wadarta vector-yada laba saf ah.
Xalka:
Ku darista laba vector oo saf ah waxaa lagu sameeyaa iyadoo lagu darayo walxaha u dhigma.
\[
\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 & 4 + 3 & 6 + 5 \end{pmatrix} = \begin{pmatrix} 3 & 7 & 11 \end{pmatrix}
\]
Markaa, wadarta guud ee \(\mathbf{a}\) iyo \(\mathbf{b}\) waa \(\begin{pmatrix} 3 & 7 & 11 \end{pmatrix}\).
Tusaale 3: Isku-dhufashada Cabbirka iyadoo loo eegayo Vektorrada Tiirarka
Su'aal:
Marka la eego vektor tiir \(\mathbf{c}\) iyo scalar \(k\):
\[
\mathbf{c} = \bilaw{pmatrix}
-3 \\
4 \\
5
\end{pmatrix}, \quad k = 2
\]
Xisaabi natiijada isku dhufashada scalar-ka.
Xalka:
Isku-dhufashada isku-dhafka ah ee vector-ka tiir waxaa lagu sameeyaa iyadoo lagu dhufanayo walax kasta oo vector ah iyadoo lagu dhufanayo scalar.
\[
k\mathbf{c} = 2 \bilaw{pmatrix}
-3 \\
4 \\
5
\dhammaad{pmatrix} = \bilaw{pmatrix}
2 jeer -3 \\
2 jeer 4
2 jeer 5
\dhammaad{pmatrix} = \bilaw{pmatrix}
-6 \\
8 \\
10
\dhammaad{pmatrix}
\]
Markaa, natiijada ku dhufashada scalar \(2\) ee vector-ka tiirka \(\mathbf{c}\) waa \(\begin{pmatrix} -6 \\ 8 \\ 10 \end{pmatrix}\).
Su'aal Tusaale ah 4: Isku-dhufashada Cabbirka iyadoo loo eegayo Vectors Safka ah
Su'aal:
Marka la eego vektor saf \(\mathbf{d}\) iyo scalar \(m\):
\[
\mathbf{d} = \begin{pmatrix} 7 & -2 & 1 \end{pmatrix}, \quad m = -3
\]
Xisaabi natiijada isku dhufashada scalar-ka.
Xalka:
Isku-dhufashada isku-dhafka ah ee vector-ka safka ah waxaa lagu sameeyaa iyadoo lagu dhufanayo walax kasta oo vector ah iyadoo lagu dhufanayo scalar.
\[
m\mathbf{d} = -3 \begin{pmatrix} 7 & -2 & 1 \end{pmatrix} = \begin{pmatrix} -3 \times 7 & -3 \times -2 & -3 \times 1 \end{pmatrix} = \begin{pmatrix} -21 & 6 & -3 \end{pmatrix}
\]
Markaa, natiijada ku dhufashada scalar \(-3\) ee vector-ka safka \(\mathbf{d}\) waa \(\begin{pmatrix} -21 & 6 & -3 \end{pmatrix}\).
Tusaale 5: Isku-dhufashada Matrix \(1 \times 3\) by \(3 \times 1\) (Vector Saf ah oo loo sameeyay Column Vector)
Su'aal:
Waxaa la siiyay vector saf ah \(\mathbf{e}\) iyo safafka tiirka \(\mathbf{f}\):
\[
\mathbf{e} = \bilow{pmatrix} 2 & -1 & 4 \dhamaadka{pmatrix}, \quad \mathbf{f} = \bilaaban{pmatrix}
5 \\
3 \\
-2
\dhammaad{pmatrix}
\]
Xisaabi natiijada labada vector.
Xalka:
Si loo sameeyo isku dhufashada matrix-ka, vektor-ka safka \(\mathbf{e}\) waxaa loola dhaqmaa sidii matrix \(1 \times 3\) ah, vektor-ka tiirka \(\mathbf{f}\) waxaa loola dhaqmaa sidii matrix \(3 \times 1\) ah. Natiijada isku dhufashadani waa scalar, waana wadarta wax soo saarka walxaha u dhigma:
\[
\mathbf{e} \mathbf{f} = \bilow{pmatrix} 2 & -1 & 4 \dhammaadka{pmatrix} \bilaaban{pmatrix}
5 \\
3 \\
-2
\end{pmatrix} = (2 \jeer 5) + (-1 \jeer 3) + (4 \jeer -2) = 10 - 3 - 8 = -1
\]
Markaa, natiijada ku dhufashada vektorka safka \(\mathbf{e}\) vektorka tiirka \(\mathbf{f}\) waa \(-1\).
Tusaale 6: Isku-dhufashada Matrix \(3 \times 1\) by \(1 \times 3\) (Vektar Tiirka Vector-ka Safka)
Su'aal:
Marka la eego vektor tiir \(\mathbf{g}\) iyo vektor saf \(\mathbf{h}\):
\[
\mathbf{g} = \bilaw{pmatrix}
1 \\
2 \\
3
\end{pmatrix}, \quad \mathbf{h} = \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}
\]
Xisaabi natiijada labada vector.
Xalka:
Isu-dhufashada matrix-ka ee vector-ka tiirka iyadoo la adeegsanayo vector saf ah waxay soo saartaa matrix (\(3 \times 1\)) oo lagu dhufto (\(1 \times 3\)) kaas oo soo saara matrix \(3 \times 3\) ah. Curiye kasta oo cusub waa natiijada curiyayaasha u dhigma:
\[
\mathbf{g} \mathbf{h} = \bilaw{pmatrix}
1 \\
2 \\
3
\end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix}
1 jeer 4 & 1 jeer 5 & 1 jeer 6 \\
2 jeer 4 & 2 jeer 5 & 2 jeer 6 \\
3 jeer 4 & 3 jeer 5 & 3 jeer 6
\dhammaad{pmatrix} = \bilaw{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12 & 15 & 18
\dhammaad{pmatrix}
\]
Markaa, natiijada ku dhufashada vektorka tiirka \(\mathbf{g}\) ee vektorka safka \(\mathbf{h}\) waa matrix-ka:
\[
\bilow{pmatrix}
4 & 5 & 6 \\
8 & 10 & 12 \\
12 & 15 & 18
\dhammaad{pmatrix}
\]
Gabagabo
Maqaalkan oo dhan, waxaan aragnay tusaalooyin dhowr ah oo ku saabsan vector-yada tiirarka iyo safka. Ku darista vector-yada tiirarka iyo safka labadaba waxaa lagu gaaraa iyadoo lagu darayo curiyayaashood u dhigma. Isku-dhufashada scalar-ka ee vector-ka waxaa sidoo kale lagu gaaraa iyadoo lagu dhufto curiye kasta oo vector-ka ah scalar-ka. Ugu dambeyntii, waxaan barannay sida loo dhufto vector-yada safka iyo tiirarka, iyadoo la soo saarayo scalar ama matrix, iyadoo ku xiran kala horreysiintooda. Barashada hawlgalladan aasaasiga ah waa mid muhiim u ah codsiyada aadka u adag ee aljabrada toosan iyo falanqaynta xogta.
