Mehlala ea Lipotso tse Buisanang ka Mareo, Mongolo, le Mefuta ea Li-Vector
Ho ngola divektara le kutlwisiso ya tsona di bohlokwa makaleng a fapaneng a saense, haholoholo fisiks le dipalo. Tshebediso e nepahetseng ya divektara e ka thusa ho sekaseka mathata le ho fumana ditharollo tse sebetsang hantle. Sengoloa sena se bua ka mantswe le ho ngola tse amanang le divektara, se di bontsha ka mehlala le ditlhaloso tse felletseng.
Mantswe a Vektha
Ho utloisisa li-vector, re tlameha ho utloisisa mantsoe a motheo pele:
1. Vektara: Palo e nang le boholo (boleng bo boholo) le tataiso. Vektara hangata e tšoantšetsoa ka litlhaku tse matla tse kang A, a, kapa letšoao la motsu ka holimo ho tsona joalo ka \(\vec{A}\).
2. Boholo (Boleng bo Boholo): Ena ke bolelele kapa boholo ba vekthara. E bontšoa ke | A | kapa \(\|\vec{A}\|\).
3. Hlooho le Mohatla: Ka setšoantšo sa litšoantšo, li-vector li bontšoa e le metsu. Sebaka seo motsu o qalang ho sona se bitsoa mohatla 'me ntlha ea ho qetela ea motsu e bitsoa hlooho.
4. Livektara tse Tšoanang: Livektara tse ts'oanang kapa tse ts'oanang.
5. Li-vector tsa Collinear: Li-vector tse lutseng moleng o le mong o otlolohileng.
6. Vektha e hlahisang liphello: Vektha e le 'ngoe e nang le phello e tšoanang le phello e kopaneng ea livektha tse peli kapa ho feta.
Mongolo oa Vector
Mongolo oa vector o na le melao e 'maloa e lokelang ho utloisisoa ho toloka le ho ngola vector ka nepo.
1. Mongolo wa Tlhaku e Tete le Motsu: Divekthara hangata di bontshwa ka ditlhaku tse tete kapa metsu. Mehlala: A , B , kapa \(\vec{A}\).
2. Li-Vector Coordinates: Li-vector tse sebakeng sa mahlakore a mabeli (2D) li hlalosoa e le \(\vec{A} = (A_x, A_y)\), ha sebakeng sa mahlakore a mararo (3D) li hlalosoa e le \(\vec{A} = (A_x, A_y, A_z)\).
3. Livekthara tsa Motheo: Sebakeng sa 2D le 3D, livekthara tsa motheo tse sebelisoang hangata ke \(\vec{i}\), \(\vec{j}\), le \(\vec{k}\), tse bolelang litaelo tsa x, y, le z, ka ho latellana.
4. Ts'ebetso ea Vektara:
– Ho eketsa: \(\vec{A} + \vec{B}\)
– Ho tlosa: \(\vec{A} – \vec{B}\)
– Ho Atisa ka Scalar: \(k\vec{A}\)
– Ho Atisa ha Matheba (sehlahisoa sa matheba): \(\vec{A} \cdot \vec{B}\)
– Ho Ata ho Tšela (sehlahisoa se tšelaneng): \(\vec{A} \times \vec{B}\)
Mefuta ea Vekthara
Mefuta e fapaneng ea li-vector e ka fumanoa ho latela moelelo le mofuta oa tsona:
1. Vektara ea Zero: Vektara e nang le boholo ba 0 'me e se na tataiso. E bontšoa ke 0 kapa \(\vec{0}\).
2. Vektara ea Yuniti: Vektara e nang le boholo ba 1. Hangata e sebelisoa ho bontša tataiso.
3. Vektara ea Boemo: Vektara e bontšang boemo ba ntlha e amanang le tšimoloho (0,0,0).
4. Li-vector tse tsamaellanang le tse sa tsamaellaneng: Li-vector tse ka lehlakoreng le le leng le tse fapaneng, empa li le moleng o tšoanang oa ketso.
5. Li-vector tsa Coplanar: Li-vector tse sebakeng se le seng.
Lipotso tsa Mehlala le Puisano
Potso ea 1: Ho Bala Boholo ba Vektheri
Boholo ba vekthara \(\vec{A} = (3, 4)\) ke bofe?
Karabo:
Ho bala boholo ba vekthara \(\vec{A}\), re sebelisa foromo:
\[\|\vec{A}\| = \sqrt{A_x^2 + A_y^2}\]
Kenya boleng ka mokhoa o latelang:
\[\|\vec{A}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
Kahoo, boholo ba vekthara \(\vec{A}\) ke 5.
Potso ea 2: Ho eketsa le ho tlosa li-vector
Ho fanoe ka livekthara tse peli \(\vec{A} = (2, 3)\) le \(\vec{B} = (1, -1)\). Bala \(\vec{A} + \vec{B}\) le \(\vec{A} – \vec{B}\).
Karabo:
Ho eketsa divekthara \(\vec{A}\) le \(\vec{B}\):
\[\vec{A} + \vec{B} = (2, 3) + (1, -1) = (2 + 1, 3 – 1) = (3, 2)\]
Ho tlosa divekthara \(\vec{A}\) le \(\vec{B}\):
\[\vec{A} – \vec{B} = (2, 3) – (1, -1) = (2 – 1, 3 – (-1)) = (1, 4)\]
Kahoo, \(\vec{A} + \vec{B} = (3, 2)\) le \(\vec{A} – \vec{B} = (1, 4)\).
Potso ea 3: Sehlahisoa sa Karolo
Bala sehlahisoa sa matheba sa livekthara tse peli \(\vec{A} = (2, 3)\) le \(\vec{B} = (1, 4)\).
Karabo:
Sehlahisoa sa matheba sa li-vector tse peli ke:
\[\vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y\]
Phetolo ea boleng:
\[\vec{A} \cdot \vec{B} = 2 \cdot 1 + 3 \cdot 4 = 2 + 12 = 14\]
Kahoo, sehlahisoa sa matheba sa \(\vec{A}\) le \(\vec{B}\) ke 14.
Potso ea 4: Sehlahisoa se kopaneng
Ho fanoe ka livekthara tse peli sebakeng sa mahlakore a mararo \(\vec{A} = (1, 2, 3)\) le \(\vec{B} = (4, 5, 6)\). Bala sehlahisoa se kopaneng \(\vec{A} \times \vec{B}\).
Karabo:
Sehlahisoa se kopaneng sa li-vector tse peli sebakeng sa mahlakore a mararo se hlalosoa e le se khethollang matrix e latelang:
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{vmatrix}
\]
Bakeng sa livekthara \(\vec{A}\) le \(\vec{B}\):
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
1 & 2 & 3 \\
4 & 5 le 6
\end{vmatrix}
\]
E baloa e le:
\[
\vec{A} \times \vec{B} = \vec{i}(2 \cdot 6 – 3 \cdot 5) – \vec{j}(1 \cdot 6 – 3 \cdot 4) + \vec{k}(1 \cdot 5 – 2 \cdot 4)
\]
\[
= \vec{i}(12 – 15) – \vec{j}(6 – 12) + \vec{k}(5 – 8)
\]
\[
= \vec{i}(-3) – \vec{j}(-6) + \vec{k}(-3)
\]
\[
= -3\vec{i} + 6\vec{j} – 3\vec{k}
\]
Kahoo, sehlahisoa se kopaneng sa \(\vec{A}\) le \(\vec{B}\) ke \(\vec{A} \times \vec{B} = (-3, 6, -3)\).
Ha ho sebetsanoa le mathata a vector, ho utloisisa mehopolo ea motheo le mantsoe ke ntlha ea mantlha ea ho qala. Sengoloa sena se ikemiselitse ho fa babali kutloisiso ea ts'ebetso e fapaneng ea vector le mefuta ea eona e fapaneng, e tla ba ea bohlokoa haholo tlhahlobong ea lipalo le ea 'mele.
