Su'aalo Tusaale ah oo Ka Hadlaya Halabuurka Isbeddelka Iyada oo La Adeegsanayo Matrices
Isbeddellada joomatarigu waa mowduuc muhiim ah oo ku jira xisaabta, gaar ahaan joomatari iyo aljabrada toosan. Isbeddelladan waxaa ka mid noqon kara turjumaad, wareegyo, milicsi, iyo ballaarin. Maqaalkan, waxaan ku baari doonnaa sida loo matali karo loona xallin karo isku-dhafka isbeddellada kala duwan iyadoo la adeegsanayo matrices. Waxaan sidoo kale bixin doonnaa tusaalooyin dhibaatooyin iyo xalal.
1. Hordhac ku saabsan Isbeddelka iyadoo la adeegsanayo Matrices
Isbeddellada joomatari waxaa lagu matali karaa matrices. Tusaale ahaan, isbeddelka wareegga, turjumaadda, milicsiga, iyo ballaarinta waxaa loo qaabeyn karaa qaabka matrix sida soo socota:
1. Turjumaad
\[
T(x, y) = \begin{pmatrix} x + a \\ y + b \end{pmatrix}
\]
2. Wareeg
\[
R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ sin\theta & \cos\theta \end{pmatrix}
\]
3. Milicsiga ku saabsan dhidibka X
\[
\text{Milicsiga X} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\]
4. Ballaarinta (ballaarinta/ballaarinta)
\[
D(k) = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}
\]
2. Halabuurka Isbeddellada leh Matrices
Halabuurka isbeddelka waa adeegsiga isku xigxiga ee laba ama in ka badan oo isbeddello ah oo loo sameeyo shay. Si loo xisaabiyo halabuurka isbeddelka iyadoo la adeegsanayo matrices, waxaan si fudud u dhufannaa matrices-ka matalaya isbeddellada.
Su'aalo iyo Doodo Tusaale ah
Soal
Marka la eego qodobka P(2, 3), hel natiijada isbeddelka soo socda:
1. Wareegga \(90^\circle\) dhanka saacadda (CW)
2. Ballaarinta iyadoo la adeegsanayo cabirka 2
3. Turjumaadda (1, -2)
Dood
1. Wareeg \(90^\circle\) CW
Shaxda loogu talagalay wareegga saacadda u socda ee \(90^\circle\):
\[
\begin{pmatrix} \cos(-90^\circ) & -\sin(-90^\circ) \\ \sin(-90^\circ) & \cos(-90^\circ) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
\]
Ku dabaqidda isbeddelka wareegga barta P:
\[
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 0 \cdot 2 + 1 \cdot 3 \\ -1 \cdot 2 + 0 \cdot 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}
\]
Barta P ka dib isbeddelka wareegga waa P'(3, -2).
2. Ballaarinta iyadoo la adeegsanayo cabirka 2
Matrix-ka ballaarinta oo leh qodobka 2aad ee cabbirka:
\[
\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}
\]
Ku dabaqidda isbeddelka ballaarinta barta P'(3, -2):
\[
\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 2 \cdot 3 + 0 \cdot -2 \\ 0 \cdot 3 + 2 \cdot -2 \end{pmatrix} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}
\]
Barta P' ka dib isbeddelka ballaarinta waa P”(6, -4).
3. Turjumaadda (1, -2)
Hawlgallada turjumaadda ee la bixiyay waa kuwan soo socda:
\[
T(x, y) = \begin{pmatrix} x + 1 \\ y – 2 \end{pmatrix}
\]
Ku dabaqidda beddelka turjumaadda barta P”(6, -4):
\[
T(6, -4) = \begin{pmatrix} 6 + 1 \\ -4 – 2 \begin{pmatrix} = \begin{pmatrix} 7 \\ -6 \bend{pmatrix}
\]
Markaa, dhibicda ugu dambeysa ka dib marka dhammaan isbeddellada la dabaqo waa P(7, -6).
3. Xisaabinta Halabuurka Isbeddelka
Su'aalo Dheeraad ah
Qodobka Q(1, 2) ee la bixiyay iyo isbeddelka soo socda:
1. Milicsiga ku saabsan dhidibka X.
2. Wareegga \(180^\circle\) dhanka saacadda (CW).
Dood
1. Milicsiga ku saabsan dhidibka X
Matrix-ka milicsiga ee ku saabsan dhidibka X:
\[
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\]
Ku dabaqidda isbeddelka milicsiga qodobka Q:
\[
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 2 \\ 0 \cdot 1 + (-1) \cdot 2 \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}
\]
Qodobka Q ka dib isbeddelka milicsiga waa Q'(1, -2).
2. Wareeg \(180^\circle\) CW
Matrix-ka wareegga \(180^\circle\) dhanka saacadda:
\[
\begin{pmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
\]
Ku dabaqidda beddelka wareegga \(180^\circle\) barta Q'(1, -2):
\[
\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ -2 \end{pmatrix} = \begin{pmatrix} -1 \cdot 1 + 0 \cdot -2 \\ 0 \cdot 1 + -1 \cdot -2 \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}
\]
Markaa, dhibicda ugu dambeysa ka dib marka dhammaan isbeddellada la dabaqo waa Q(-1, 2).
Xiritaanka
Habka isku-dhafka isbeddelka iyadoo la adeegsanayo matrices-ka ayaa aad waxtar ugu leh fududeynta iyo xisaabinta isbeddellada joomatari. Markaan raacno tallaabooyinka kor ku xusan, si fudud ayaan u fahmi karnaa oo u dabaqi karnaa noocyada kala duwan ee isbeddellada hal dhibic ama shay kale oo joomatari ah. Barashada isticmaalka matrices-ka ee isbeddellada ayaa sidoo kale sahlaysa in lagu dabaqo meelo kala duwan sida fiisikiska, sawirada kombiyuutarka, iyo waxyaabo kaloo badan.
