Piv txwv lus nug tham txog kev ntxiv ob lub vectors siv txoj kev parallelogram
Kev ntxiv vector yog ib lub tswv yim tseem ceeb hauv physics thiab lej, feem ntau siv los piav qhia txog tej xwm txheej ntuj tsim thiab teeb meem hauv lub neej txhua hnub. Muaj ntau txoj hauv kev los ntxiv ob lub vectors, ib qho yog txoj kev parallelogram. Txoj kev no tsis yog tsuas yog yooj yim xwb tab sis kuj muab kev pom zoo ntawm ob lub vectors sib koom ua ke los ua ib lub vector tshwm sim. Hauv tsab xov xwm no, peb yuav saib ntau qhov piv txwv ntawm kev ntxiv vector siv txoj kev parallelogram, nrog rau lawv cov kev daws teeb meem.
Vector yog dab tsi?
Ua ntej peb nkag mus rau hauv cov teeb meem piv txwv, peb yuav tsum nkag siab txog lub ntsiab lus ntawm vector. Ib vector yog ib qho ntau uas muaj ob qho tib si qhov loj (ntev) thiab kev coj. Cov piv txwv ntawm vectors suav nrog qhov ceev, kev ua kom nrawm, lub zog, thiab kev hloov chaw. Ib vector tuaj yeem sawv cev ua nws cov khoom (i, j, k) hauv Cartesian coordinates lossis ua nws qhov ntev thiab kev coj (lub kaum sab xis).
Txoj Kev Siv Parallelogram
Txoj kev parallelogram yog ib txoj hauv kev los ntxiv ob lub vectors. Hauv txoj kev no, peb sawv cev rau ob lub vectors ua ob sab ntawm lub parallelogram. Lub vector tshwm sim yog lub diagonal ntawm lub parallelogram pib ntawm qhov pib ntawm ob lub vectors. Hauv lej, yog tias peb muaj ob lub vectors \(\vec{A}\) thiab \(\vec{B}\), qhov tshwm sim yog \(\vec{R} = \vec{A} + \vec{B} \).
Cov kauj ruam ib kauj ruam rau kev siv txoj kev parallelogram yog raws li nram no:
1. Kos daim vector \(\vec{A}\) ntawm qhov chaw pib.
2. Los ntawm qhov kawg ntawm lub vector \(\vec{A}\), kos lub vector \(\vec{B}\).
3. Kos ib txoj kab sib luag rau lub vector \(\vec{B}\) los ntawm qhov chaw pib \(\vec{A}\).
4. Kos ib txoj kab sib luag rau vector \(\vec{A}\) los ntawm qhov kawg ntawm vector \(\vec{B}\).
5. Kos ib txoj kab pheeb ces kaum ntawm qhov chaw pib mus rau lub ces kaum sab nraud kom tau txais qhov vector \(\vec{R}\).
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Lo lus nug thib 1
Xav tias peb muaj ob lub vectors \(\vec{A}\) thiab \(\vec{B}\):
- \(\vec{A}\) muaj qhov ntev (qhov loj) ntawm 5 units thiab qhov kev taw qhia ntawm 0° (lossis raws txoj kab x zoo),
- \(\vec{B}\) muaj qhov ntev ntawm 3 units thiab qhov kev taw qhia ntawm 90° (lossis raws txoj kab y zoo).
Tus nqi ntawm kev ntxiv ob lub vectors no siv txoj kev parallelogram yog dab tsi?
Kev Sib Tham:
1. Kos tus vector \(\vec{A}\) raws txoj kab x zoo nrog qhov ntev ntawm 5 units.
2. Los ntawm qhov kawg ntawm lub vector \(\vec{A}\), kos lub vector \(\vec{B}\) raws txoj kab y zoo nrog qhov ntev ntawm 3 units.
3. Txij ntawm qhov chaw pib \(\vec{A}\), kos ib txoj kab sib luag rau \(\vec{B}\).
4. Txij thaum kawg ntawm \(\vec{B}\), kos ib txoj kab sib luag rau \(\vec{A}\).
5. Qhov tshwm sim yog ib daim duab parallelogram nrog ib daim diagonal uas yog qhov tshwm sim vector \(\vec{R}\).
Vim tias \(\vec{A}\) thiab \(\vec{B}\) yog perpendicular rau ib leeg, peb tuaj yeem siv Pythagorean theorem los xam qhov ntev ntawm cov vector tshwm sim:
\[ R = \sqrt{A^2 + B^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \kwv yees li 5.83 \]
Qhov kev taw qhia ntawm cov vector tshwm sim tuaj yeem suav tau los ntawm kev siv trigonometry. Yog tias \(\theta\) yog lub kaum sab xis ntawm cov tshwm sim thiab \(\vec{A}\):
\[ \tan(\theta) = \frac{B}{A} = \frac{3}{5} \]
yog li ntawd:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \approx 30.96^\circ \]
Yog li, qhov tshwm sim vector \(\vec{R}\) muaj qhov loj ntawm kwv yees li 5.83 units thiab kev taw qhia ntawm kwv yees li 30.96 ° ntawm \(\vec{A}\).
Lo lus nug thib 2
Ob lub vectors \(\vec{C}\) thiab \(\vec{D}\) raug muab raws li nram no:
- \(\vec{C}\) nrog qhov ntev ntawm 4 units thiab kev taw qhia ntawm 45°.
- \(\vec{D}\) nrog qhov ntev ntawm 6 units thiab kev taw qhia ntawm 120°.
Txheeb xyuas qhov vector uas tau los ntawm kev ntxiv ob lub vectors.
Kev Sib Tham:
Yuav ntxiv ob lub vectors uas tsis sib ncaj rau ib leeg lossis nyob rau hauv cov duab sib txawv, koj tuaj yeem siv Cartesian Cheebtsam.
1. Rhuav \(\vec{C}\) thiab \(\vec{D}\) ua tej yam x thiab y.
Rau \(\vec{C}\):
\[ C_x = C \cos(45^\circ) = 4 \cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \]
\[ C_y = C \sin(45^\circ) = 4 \sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \]
Rau \(\vec{D}\):
\[ D_x = D \cos(120^\circ) = 6 \cos(120^\circ) = 6 \cdot (-\frac{1}{2}) = -3 \]
\[ D_y = D \sin(120^\circ) = 6 \sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \approx 5.20 \]
2. Ntxiv cov x thiab y Cheebtsam ntawm ob qho vectors:
\[ R_x = C_x + D_x = 2.83 + (-3) = -0.17 \]
\[ R_y = C_y + D_y = 2.83 + 5.20 = 8.03 \]
3. Xam qhov loj thiab kev coj ntawm qhov vector \(\vec{R}\):
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.03 + 64.48} = \sqrt{64.51} \kwv yees li 8.03 \]
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{8.03}{-0.17}\right) \approx \tan^{-1}(-47.24) \]
Vim tias qhov tshwm sim tsis zoo, peb ntxiv 180 ° kom tau lub kaum sab xis hauv qhov system quadrant kom raug:
\[ \theta \approx \tan^{-1}(47.24) + 180^\circ \approx 271.93^\circ \]
Yog li, qhov tshwm sim vector \(\vec{R}\) muaj qhov loj li ntawm 8.03 units thiab kev taw qhia li ntawm 271.93°, lossis peb tuaj yeem hais li ntawm 91.93° los ntawm qhov tsis zoo x-axis hauv plaub quadrant.
Kev kaw
Txoj kev parallelogram yog ib txoj hauv kev zoo thiab pom tseeb los ntxiv ob lub vectors. Txawm hais tias txoj kev no yuav zoo li yooj yim rau cov vectors yooj yim, nws yog ib qho tseem ceeb kom nkag siab tias rau cov vectors nyuaj dua, peb feem ntau yuav tsum siv cov khoom Cartesian thiab cov txheej txheem algebraic siab dua kom tau txais cov txiaj ntsig raug. Vam tias, cov piv txwv saum toj no muab ib daim duab meej ntawm yuav ua li cas txoj kev no tuaj yeem siv rau hauv ntau qhov xwm txheej.
