Liveketara tsa Kholomo le Liveketara tsa Mela: Metheo ea Lipalo le Ts'ebeliso ea Tsona
Lipalong le saenseng, khopolo ea li-vector ke khopolo ea motheo. Li-vector li sebelisoa ho emela bongata bo nang le tataiso le boholo. Ntle le tšebeliso ea tsona lipalong, li-vector li boetse li fumana lits'ebetso mafapheng a fapaneng a kang fisiks, boenjiniere le litšoantšo tsa khomphutha. Moelelong oa algebra e otlolohileng, li-vector hangata li arotsoe ka mefuta e 'meli e meholo: li-vector tsa kholomo le li-vector tsa mola. Sengoloa sena se tla hlahloba likhopolo tsa li-vector tsa kholomo le li-vector tsa mola ka botebo, hammoho le ts'ebeliso ea tsona masimong a fapaneng.
Litlhaloso le Litlhaloso
Vektheri ea Kholomo
Vekthara ea kholomo ke vekthara e emeloang e le kholomo e otlolohileng. Mongolo o akaretsang oa vekthara ea kholomo ke o latelang:
\[
\mathbf{v} = \begin{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\ pheletso{bmatrix}
\]
Moo \(v_1, v_2, \ldots, v_n\) e leng dielemente tsa vekthara. Palo ya dielemente tse ka hara vekthara e bontsha boholo ba vekthara.
Vekthara ea Mola
Ka lehlakoreng le leng, vekthara ea mola ke vekthara e emeloang e le mola o rapameng. Mongolo o akaretsang oa vekthara ea mola ke o latelang:
\[
\mathbf{u} = \begin{bmatrix}
u_1 le u_2 le \cdots le u_n
\ pheletso{bmatrix}
\]
Jwalo ka vektara ya kholomo, \(u_1, u_2, \ldots, u_n\) ke dielemente tsa vektara hammoho le boholo ba vektara.
Mesebetsi ea Motheo ka Li-vector tsa Kholomo le Li-vector tsa Mela
Ho eketsa le ho tlosa
Livekthara tsa kholomo le livekthara tsa mola ka bobeli li ka eketsoa le ho tlosoa haeba li na le litekanyo tse tšoanang. Mohlala, bakeng sa livekthara tse peli tsa kholomo \(\mathbf{v}\) le \(\mathbf{w}\) tse nang le likarolo \(v_i\) le \(w_i\), ka ho latellana, tlatsetso ke:
\[
\mathbf{v} + \mathbf{w} = \qala{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\end{bmatrix} + \begin{bmatrix}
w_1 \\
w_2 \\
\vdots \\
w_n
\end{bmatrix} = \begin{bmatrix}
v_1 + w_1 \\
v_2 + w_2 \\
\vdots \\
v_n + w_n
\ pheletso{bmatrix}
\]
Ha e le li-vector tsa mela, molao-motheo o ts'oana:
\[
\mathbf{u} + \mathbf{t} = \begin{bmatrix}
u_1 le u_2 le \cdots le u_n
\end{bmatrix} + \begin{bmatrix}
t_1 & t_2 & \cdots & t_n
\end{bmatrix} = \begin{bmatrix}
u_1 + t_1 & u_2 + t_2 & \cdots & u_n + t_n
\ pheletso{bmatrix}
\]
Ho Atisa ha Scalar
Ho atisa ka scalar ho kenyelletsa ho atisa karolo e 'ngoe le e 'ngoe ea vector ka nomoro ea scalar. Mohlala, haeba scalar \(c\) le vector ea kholomo \(\mathbf{v}\), joale:
\[
c\mathbf{v} = c \qala{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\end{bmatrix} = \begin{bmatrix}
cv_1 \\
cv_2 \\
\vdots \\
cv_n
\ pheletso{bmatrix}
\]
'Me haeba vekthara ea mola \(\mathbf{u}\):
\[
c\mathbf{u} = c \ qala{bmatrix}
u_1 le u_2 le \cdots le u_n
\end{bmatrix} = \begin{bmatrix}
cu_1 & cu_2 & \cdots & cu_n
\ pheletso{bmatrix}
\]
Ho Atisa ha Vektara
Ho ata ha vector ho kenyelletsa mefuta e mengata ho tloha sehlahisoa sa matheba ho ea sehlahisoa se kopaneng.
Bakeng sa divekthara tse pedi tsa kholomo \(\mathbf{v}\) le \(\mathbf{w}\), sehlahiswa sa matheba se hlahiswa ka tsela ena:
\[
\mathbf{v} \cdot \mathbf{w} = \ kakaretso_{i=1}^n v_i w_i
\]
Sephetho sa sehlahisoa sa matheba ke scalar. Leha ho le jwalo, sehlahisoa se tshekaletseng se hlaloswa feela bakeng sa divekthara tse sebakeng sa mahlakore a mararo mme se hlahisa vekthara e ntjha e tsamaellanang le divekthara ka bobedi tsa mantlha.
Likopo Masimong a sa Tšoaneng
Fisiks
Fisiks, livekthara tsa kholomo le livekthara tsa mola hangata li sebelisoa ho emela bongata bo fapaneng ba 'mele joalo ka lebelo, ho potlakisa le masimo a matla. Mohlala, ho potlakisa ha matla a khoheli sebakeng se itseng ho ka emeloa e le vekthara ea kholomo ea mahlakore a mararo:
\[
\mathbf{a} = \begin{bmatrix}
0 \\
-9.8 \\
0
\end{bmatrix} \, \text{m/s}^2
\]
Boenjiniere le Theknoloji
Boenjiniere, haholo-holo tlhahlobong ea sebopeho, li-vector tsa kholomo hangata li sebelisoa ho emela matla le linako tse meahong. Mohlala, matla a libakeng tsa khokahano sebopehong sa foreimi a ka emeloa e le li-vector tsa kholomo:
\[
\mathbf{F} = \begin{bmatrix}
F_x \\
F_y \\
F_z
\ pheletso{bmatrix}
\]
Moo \(F_x, F_y,\) le \(F_z\) e leng dikarolo tsa matla ka ditsela tse tharo tse otlolohileng.
Saense ea Khomphutha le Litšoantšo tsa Khomphutha
Ho k'homphieutha, li-vector lia hlokahala bakeng sa ho emela le ho laola data. Litšoantšong tsa khomphutha, li-vector li sebelisoa ho emela lintlha, li-vector tsa boemo le liphetoho. Mohlala, ntlha sebakeng sa mahlakore a mararo e ka emeloa e le vector ea kholomo:
\[
\mathbf{p} = \begin{bmatrix}
x \\
y \\
z
\ pheletso{bmatrix}
\]
Liphetoho tse kang liphetolelo, lipotoloho le likala le tsona li emeloa ka mokhoa o kopaneng ho sebelisoa matrices e sebetsang ho li-vector tsa kholomo kapa tsa mola.
Mekhoa ea ho Rarolla Li-equation tse Kholo
Hangata li-vector tsa kholomo le li-vector tsa mela li sebelisoa ho rarolla litsamaiso tsa li-equation tse otlolohileng. Mohlala, sistimi e latelang ea li-equation tse otlolohileng:
\[
\ qala{maemo}
a_{11}x_1 + a_{12}x_2 = b_1 \\
a_{21}x_1 + a_{22}x_2 = b_2
\qetela{maemo}
\]
E ka emeloa ka mokhoa oa matrix ka tsela e latelang:
\[
\begin{bmatrix}
a_{11} le a_{12} \\
a_{21} le a_{22}
\ pheletso{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\ pheletso{bmatrix}
=
\begin{bmatrix}
b_1 \\
b_2
\ pheletso{bmatrix}
\]
Mokhoa ona o etsa hore ho be bonolo haholo ho sebelisa mekhoa ea algebra e otlolohileng joalo ka ho felisa Gaussian, ho arohana ha LU, kapa esita le mekhoa ea ho pheta-pheta bakeng sa litsamaiso tse rarahaneng haholoanyane.
Qetello
Li-vector tsa kholomo le li-vector tsa mola ke lintho tsa motheo tse atisang ho bonahala li le bonolo empa li na le lits'ebetso tse kholo mafapheng a fapaneng a saense le boenjiniere. Ho utloisisa metheo ea ts'ebetso ea li-vector ke mohato oa pele oa bohlokoa oa ho tseba algebra e otlolohileng le lithuto tse ling tsa lipalo. Ka bobeli li fana ka litsela tse sebetsang tsa ho emela le ho laola data masimong a fapaneng, ho tloha fisiks le boenjiniere ho isa saenseng ea khomphutha. Kutloisiso e tebileng ea li-vector tsa kholomo le li-vector tsa mola e ka bula tsela bakeng sa likhopolo tse rarahaneng le lits'ebetso tsa lefats'e la nnete.
