Tlhahlobo ea Vector Sebakeng
Tlhahlobo ea li-vector sebakeng ke lekala la lipalo le shebaneng le thuto ea li-vector le ts'ebetso ea tsona sebakeng sa mahlakore a mararo (3D). Vector ke bongata bo nang le boholo le tataiso, ho fapana le scalar, e nang le boholo feela. Li-vector sebakeng li sebelisoa lithutong tse fapaneng, ho tloha fisiks ho ea saenseng ea khomphutha, 'me ke lisebelisoa tsa bohlokoa tlhahlobong ea jeometri, kinematics le dynamics.
Khopolo ea Motheo ea Li-vector
Vekthara sebakeng sa mahlakore a mararo e ka hlaloswa e le v = (v₁, v₂, v₃), moo v₁, v₂, le v₃ e leng dikarolo tsa vekthara ka mahlakoreng a x, y, le z, ka ho latellana. Setšoantšo sa vekthara ke motsu o nkiloeng ho tloha tšimolohong (0, 0, 0) ho ea ntlheng (v₁, v₂, v₃). Bolelele ba vekthara (boholo) bo ka baloa ho sebelisoa foromo:
\[\| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]
Ts'ebetso ea Motheo ho Li-vector
1. Ho eketsa le ho tlosa
Li-vector tse peli u = (u₁, u₂, u₃) le v = (v₁, v₂, v₃) li ka eketsoa kapa tsa tlosoa ka ho eketsa kapa ho ntša likarolo tsa tsona:
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3) \]
\[ \mathbf{u} – \mathbf{v} = (u_1 – v_1, u_2 – v_2, u_3 – v_3) \]
2. Ho atisoa ka Scalar
Haeba c ke scalar (nomoro ea 'nete), joale katoloso ea vector v ka scalar c ke:
\[ c\mathbf{v} = (cv_1, cv_2, cv_3) \]
3. Sehlahisoa sa Dot
Sehlahisoa sa matheba pakeng tsa livekthara tse peli u le v ke scalar e hlalosoang e le:
\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \]
Sehlahisoa sena sa matheba se boetse se bolela hore na livekthara tse peli li bapile, hobane livekthara tse peli tse otlolohileng (tse otlolohileng) li na le sehlahisoa sa matheba se lekanang le lefela.
4. Sehlahisoa se fapaneng
Sehlahisoa se kopaneng sa livekthara tse peli u le v se hlahisa vekthara e ncha e lumellanang le tsona ka bobeli. E hlalosoa ka tsela ena:
\[ \mathbf{u} \times \mathbf{v} = \left( u_2v_3 – u_3v_2, u_3v_1 – u_1v_3, u_1v_2 – u_2v_1 \right) \]
Likopo tsa Tlhahlobo ea Vector
1. Kinematics
Ho kinematics, motsamao oa ntho o hlalosoa ho sebelisoa li-vector tsa boemo, lebelo le ho potlakisa. Mohlala, haeba ntho e tsamaea sebakeng sa 3D, boemo ba eona ka nako t bo ka hlalosoa ke vector ea boemo r(t). Lebelo la ntho ke derivative ea vector ea boemo mabapi le nako:
\[ \mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} \]
Leha ho potlakisa e le derivative ea velocity velocity:
\[ \mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt} \]
2. Matla a ho Fetola
Ho dynamics, tlhahlobo ea vector hangata e sebelisoa ho bala matla a sebetsang nthong. Mohlala, molao oa bobeli oa Newton o ka hlalosoa ka mokhoa oa vector e le:
\[ \mathbf{F} = m\mathbf{a} \]
moo F e leng matla a sebele a sebetsang nthong ka boima m, mme a ke ho potlakisa ha ntho.
3. Matla a motlakase
Matla a motlakase a boetse a sebelisa tlhahlobo ea vector haholo. Mohlala, tšimo ea motlakase E le matla a makenete B ka bobeli ke li-vector tse itšetlehileng ka boemo ba tsona sebakeng. Li-equation tsa Maxwell, tse hlalosang kamoo masimo a motlakase le makenete a fetohang kateng, ke li-equation tse fapaneng ka sebopeho sa vector.
4. Litšoantšo tsa Khomphutha
Litšoantšong tsa khomphutha le animation, li-vector li sebelisoa ho emela boemo, tataiso le boholo ba lintho tse sebakeng sa mahlakore a mararo. Liphetoho tsa jeometri tse kang phetolelo, potoloho le boholo li sebelisoa linthong tsena ho sebelisoa matrices ea phetoho e sebetsang holim'a li-vector tsa boemo ba lintlha tsa ntho.
Phetoho ea Linear
Phetoho e otlolohileng ke mosebetsi o hokahanyang vekthara le vekthara e 'ngoe sebakeng se le seng, ka mokhoa o otlolohileng. Phetoho ena e ka emeloa ke matrix. A re re T ke phetoho e otlolohileng 'me A ke matrix ea eona. Haeba v ke vekthara, joale phetoho e otlolohileng e ka ngoloa ka tsela ena:
\[ T(\mathbf{v}) = \mathbf{A} \mathbf{v} \]
Liphetoho tse otlolohileng li kenyelletsa ho potoloha, ho bonahatsa, ho atoloha le ho kuta.
Matrix ea Phetoho
Phetoho e 'ngoe le e 'ngoe e otlolohileng e ka emeloa ke matrix. Mehlala e meng ea matrix ea phetoho ke ena:
1. Ho potoloha
Ho potoloha ho potoloha mothapo wa z ka sekhutlo θ ho hlahiswa ke matrix:
\[
\mathbf{R}_z(\theta) = \qala{pmatrix}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 le 1
\end{pmatrix}
\]
2. Ho nahanisisa
Ho bonahatsa sebopeho sa xy ho hlahiswa ke matrix:
\[
\mathbf{R}_{xy} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 le 0 le -1
\end{pmatrix}
\]
3. Sekala
Phetoho ea sekala ka ntlha ea s mahlakoreng 'ohle (isotropic) e hlahisoa ke matrix:
\[
\mathbf{S}(s) = \begin{pmatrix}
s & 0 & 0 \\
0 le s le 0 \\
0 le 0 le s
\end{pmatrix}
\]
Li-Eigenvector le Litekanyetso tsa Eigen
Moelelong oa liphetoho tse otlolohileng, li-eigenvector le li-eigenvalues ke likhopolo tsa bohlokoa. A re re A ke matrix ea phetoho e otlolohileng, λ ke eigenvalue 'me v ke eigenvector, joale:
\[ \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \]
Eigenvector ke vekthara eo tataiso le sekala sa yona di bolokilweng ka mora phetoho, ha eigenvalue e le ntlha ya sekala seo. Tlhahlobo ya eigenvectors le eigenvalues e re dumella ho utlwisisa thepa ya matrices le diphetoho tse rarahaneng tsa mola.
Qetello
Tlhahlobo ea li-vector ke sesebelisoa se matla le se sebetsang ka bongata lipalo le mahlale. Ka ho utloisisa ts'ebetso ea mantlha ea li-vector le lits'ebetso tsa tsona, re ka rarolla mathata a fapaneng fisiks, boenjiniere, litšoantšo tsa khomphutha le masimo a mang a mangata. Ho tseba likhopolo tsa liphetoho tse otlolohileng, lihlahisoa tsa matheba, lihlahisoa tse tšekaletseng, le li-eigenvector le li-eigenvalues ho re lumella ho sekaseka le ho etsa mohlala oa litsamaiso tse rarahaneng haholo ka katleho le ka botlalo.
