Mohlala oa Potso e Buisanang ka ho Eketsoa ha Li-vector tse peli ho sebelisoa Mokhoa oa Parallelogram
Ho eketsa vector ke mohopolo oa bohlokoa fisiks le lipalo, hangata o sebelisoang ho hlalosa liketsahalo tsa tlhaho le mathata a bophelo ba letsatsi le letsatsi. Ho na le mekhoa e 'maloa ea ho eketsa vector tse peli, e 'ngoe ea tsona ke mokhoa oa parallelogram. Mokhoa ona ha o utloahale feela empa o boetse o fana ka pono e matla ea kamoo vector tse peli li kopanang ho etsa vector e hlahang. Sehloohong sena, re tla sheba mehlala e 'maloa ea ho eketsa vector ho sebelisa mokhoa oa parallelogram, hammoho le litharollo tsa tsona.
Vekthara ke eng?
Pele re kena mathateng a mohlala, re hloka ho utloisisa tlhaloso ea motheo ea vector. Vector ke bongata bo nang le boholo (bolelele) le tataiso. Mehlala ea khale ea li-vector e kenyelletsa lebelo, ho potlakisa, matla le ho falla. Vector e ka emeloa e le likarolo tsa eona (i, j, k) ho li-coordinate tsa Cartesian kapa e le bolelele le tataiso ea eona (angle).
Mokhoa oa Paralelogram
Mokhoa oa parallelogram ke tsela e 'ngoe ea ho eketsa li-vector tse peli. Mokhoeng ona, re emela li-vector tse peli e le mahlakore a mabeli a parallelogram. Vector e hlahang ke daegonale ea parallelogram e qalang ho tloha moo li-vector tse peli li qalang teng. Ho ea ka lipalo, haeba re na le li-vector tse peli \(\vec{A}\) le \(\vec{B}\), sephetho ke \( \vec{R} = \vec{A} + \vec{B} \).
Mokhoa oa mohato ka mohato oa ho sebelisa mokhoa oa parallelogram ke o latelang:
1. Thala vekthara \(\vec{A}\) ho tloha moo e qalang teng.
2. Ho tloha pheletsong ea vekthara \(\vec{A}\), taka vekthara \(\vec{B}\).
3. Thala mola o bapileng le vekthara \(\vec{B}\) ho tloha moo ho qalwang teng \(\vec{A}\).
4. Thala mola o bapileng le vekthara \(\vec{A}\) ho tloha pheletsong ya vekthara \(\vec{B}\).
5. Thala daegonale ho tloha ntlha ya qalo ho ya sekhutlong se ka lehlakoreng le leng ho fumana vekthara e hlahang \(\vec{R}\).
Lipotso tsa Mehlala le Puisano
Potso ea 1
A re re re na le livekthara tse peli \(\vec{A}\) le \(\vec{B}\):
– \(\vec{A}\) e na le bolelele (boholo) ba diyuniti tse 5 le tataiso ya 0° (kapa ho latela x-axis e ntle),
– \(\vec{B}\) e na le bolelele ba diyuniti tse 3 le tataiso ya 90° (kapa ho latela mothapo o motle wa y).
Boleng ba sephetho sa ho eketsa divekthara tsena tse pedi ho sebediswa mokgwa wa parallelogram ke bofe?
Puisano:
1. Thala vekthara \(\vec{A}\) ho latela positive x-axis ka bolelele ba diyuniti tse 5.
2. Ho tloha pheletsong ea vekthara \(\vec{A}\), taka vekthara \(\vec{B}\) ho latela mothapo o motle oa y o nang le bolelele ba liyuniti tse 3.
3. Ho tloha moo ho qalwang teng \(\vec{A}\), taka mola o bapileng le \(\vec{B}\).
4. Ho tloha pheletsong ya \(\vec{B}\), taka mola o bapileng le \(\vec{A}\).
5. Sephetho ke parallelogram e nang le daegonale e leng vekthara e hlahang \(\vec{R}\).
Kaha \(\vec{A}\) le \(\vec{B}\) di shebane ka ho otloloha, re ka sebedisa theorem ya Pythagorean ho bala bolelele ba vektha e hlahang:
\[ R = \sqrt{A^2 + B^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \hoo e ka bang 5.83 \]
Tsela eo vekthara e hlahang ka yona e ka balwa ka ho sebedisa trigonometry. Haeba \(\theta\) e le sekhutlo se pakeng tsa se hlahang le \(\vec{A}\):
\[ \tan(\theta) = \frac{B}{A} = \frac{3}{5} \]
kahoo:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \hoo e ka bang 30.96^\circ \]
Ka hona, vekthara e hlahang \(\vec{R}\) e na le boholo ba diyuniti tse ka bang 5.83 le tataiso ya hoo e ka bang 30.96° ho tloha \(\vec{A}\).
Potso ea 2
Li-vector tse peli \(\vec{C}\) le \(\vec{D}\) li fanoe ka tsela e latelang:
– \(\vec{C}\) e bolelele ba diyuniti tse 4 le tataiso ya 45°.
– \(\vec{D}\) e bolelele ba diyuniti tse 6 le tataiso ya 120°.
Fumana vekthara e hlahang \(\vec{R}\) ho tsoa ho kenyelletsong ea livekthara tse peli.
Puisano:
Ho eketsa li-vector tse peli tse sa shebaneng kapa tse sa tšoaneng, u ka sebelisa likarolo tsa Cartesian.
1. Arola \(\vec{C}\) le \(\vec{D}\) ka likarolo tsa x le y.
Bakeng sa \(\vec{C}\):
\[ C_x = C \cos(45^\circ) = 4 \cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \hoo e ka bang 2.83 \]
\[ C_y = C \sin(45^\circ) = 4 \sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \hoo e ka bang 2.83 \]
Bakeng sa \(\vec{D}\):
\[ D_x = D \cos(120^\circ) = 6 \cos(120^\circ) = 6 \cdot (-\frac{1}{2}) = -3 \]
\[ D_y = D \sin(120^\circ) = 6 \sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \hoo e ka bang 5.20 \]
2. Kenya likarolo tsa x le y tsa livekthara ka bobeli:
\[ R_x = C_x + D_x = 2.83 + (-3) = -0.17 \]
\[ R_y = C_y + D_y = 2.83 + 5.20 = 8.03 \]
3. Bala boholo le tataiso ea vektha e hlahang \(\vec{R}\):
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.03 + 64.48} = \sqrt{64.51} \hoo e ka bang 8.03 \]
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{8.03}{-0.17}\right) \approx \tan^{-1}(-47.24) \]
Kaha sephetho se le mpe, re eketsa 180° ho fumana sekhutlo tsamaisong e nepahetseng ea quadrant:
\[ \theta \approx \tan^{-1}(47.24) + 180^\circ \approx 271.93^\circ \]
Kahoo, vekthara e hlahang \(\vec{R}\) e na le boholo ba diyuniti tse ka bang 8.03 le tataiso e ka bang 271.93°, kapa re ka re hoo e ka bang 91.93° ho tloha ho x-axis e mpe karolong ya bone ya dikhutlotharo.
Ho koala
Mokhoa oa parallelogram ke mokhoa o atlehang le o bonahalang oa ho eketsa li-vector tse peli. Le hoja mokhoa ona o ka bonahala o le bonolo bakeng sa li-vector tse bonolo, ho bohlokoa ho utloisisa hore bakeng sa li-vector tse rarahaneng haholoanyane, hangata re hloka ho sebelisa likarolo tsa Cartesian le mekhoa e tsoetseng pele ea algebraic ho fumana liphetho tse nepahetseng. Re tšepa hore mehlala e kaholimo e fana ka setšoantšo se hlakileng sa hore na mokhoa ona o ka sebelisoa joang maemong a fapaneng.
