Mibvunzo Yemuenzaniso Inotaura Nezvekuwedzera Vector neComponent
Kuwedzerwa kwevector inzira inokosha mufizikisi nemasvomhu inoshandiswa kuwana mhedzisiro yemavector maviri kana anopfuura. Nzira yekugadzirisa kuwedzera kwevector maererano nezvikamu inzira inobatsira zvikuru, kunyanya pakubata nemavector muzvikamu zviviri kana zvitatu. Chinyorwa chino chichatsanangura pfungwa yekuwedzera kwevector maererano nezvikamu uye kupa mienzaniso yakawanda yezvinetso nemhinduro.
Pfungwa yekuwedzera kweComponential Vector
Vector yega yega iri munzvimbo ine mativi maviri (2D) inogona kupatsanurwa kuita zvikamu zviviri: chikamu che x (horizontal) uye chikamu che y (vertical). Muzvikamu zvitatu (3D), vectors vane chimwe chikamu, chikamu che z (depth).
Ngatitii tine mavector maviri A na B. Zvikamu zvemavector aya zvinogona kuratidzwa seizvi:
– Vekitori A ine zvikamu \(A_x\) uye \(A_y\) mu2D (kana zvakare \(A_z\) mu3D).
– Vekitori B ine zvikamu \(B_x\) uye \(B_y\) mu2D (kana zvakare \(B_z\) mu3D).
Kuwedzerwa kwemavector maviri aya kuchaburitsa vector R ine zvikamu zvinotevera:
\[ R_x = A_x + B_x \]
\[ R_y = A_y + B_y \]
Kune mavectors mu3D, chikamu che z ndeichi chinoteverawo:
\[ R_z = A_z + B_z \]
Mushure mekuverenga chikamu chimwe nechimwe chevector inobuda, tinogona kuwana modulus (hukuru) uye gwara revector inobuda tichishandisa fomura:
\[ |R| = \sqrt{R_x^2 + R_y^2} \] (ye2D)
Kana ye3D:
\[ |R| = \sqrt{R_x^2 + R_y^2 + R_z^2} \]
Uye kutungamira kwevector inobuda kunogona kutsanangurwa nekona kuenda kuma coordinate axes.
Mibvunzo yemuenzaniso nekukurukurirana
Mubvunzo 1
Zvichipiwa mavector maviri mundege ine mativi maviri:
– A iri \(5 \, \text{unit}\) kumabvazuva.
– B iri \(3 \, \text{unit}\) kuchamhembe.
Sarudza vhekitari inobuda R.
Kukurukurirana
Kutanga, tinoshandura vector kuita zvikamu zvayo.
– Vekitori A: \(A = (5, 0)\) nekuti ine chikamu che x chete.
– Vekitori B : \(B = (0, 3)\) nekuti ine chikamu che y chete.
Heino huwandu hwezvikamu:
\[ R_x = A_x + B_x = 5 + 0 = 5 \]
\[ R_y = A_y + B_y = 0 + 3 = 3 \]
Ipapo vhekitori inobuda R ndeiyi:
\[ R = (5, 3) \]
Kuti uverenge urefu (modulus) yevector R:
\[ |R| = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \inenge 5.83 \]
Kutungamira kwevector R kunogona kuverengerwa uchishandisa kona θ kuenda ku x-axis:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{3}{5} \]
\[ \theta = \arctan\left(\frac{3}{5}\right) \approx 30.96^\circ \]
Saka, vhekita yeR inobuda ine urefu hwemayuniti angangoita 5.83 uye inoumba kona ye30.96° ne x-axis.
Mubvunzo 2
Zvichipiwa mavector maviri muzvikamu zvitatu:
– A ndi \(3\hat{i} + 2\hat{j} + 1\hat{k}\)
– B ndiye \(1\hat{i} + 4\hat{j} + 2\hat{k}\)
Sarudza vhekitari inobuda R.
Kukurukurirana
Kutanga, tinoona zvikamu zvevector yega yega:
– Vekitori A: \(A_x = 3\), \(A_y = 2\), \(A_z = 1\).
– Vekitori B : \(B_x = 1\), \(B_y = 4\), \(B_z = 2\).
Heino huwandu hwezvikamu:
\[ R_x = A_x + B_x = 3 + 1 = 4 \]
\[ R_y = A_y + B_y = 2 + 4 = 6 \]
\[ R_z = A_z + B_z = 1 + 2 = 3 \]
Ipapo vhekitori inobuda R ndeiyi:
\[ R = (4, 6, 3) \]
Kuti uverenge urefu (modulus) yevector R:
\[ |R| = \sqrt{4^2 + 6^2 + 3^2} = \sqrt{16 + 36 + 9} = \sqrt{61} \inenge 7.81 \]
Kutungamira kwevector R maererano ne x, y, uye z axes kunogona kuverengerwa uchishandisa cosine yedirector:
\[ \cos(\alpha) = \frac{R_x}{|R|} = \frac{4}{7.81} \inenge 0.512 \]
\[ \alpha = \arccos(0.512) \inenge 59.50^\circ \]
\[ \cos(\beta) = \frac{R_y}{|R|} = \frac{6}{7.81} \inenge 0.768 \]
\[ \beta = \arccos(0.768) \inenge 39.50^\circ \]
\[ \cos(\gamma) = \frac{R_z}{|R|} = \frac{3}{7.81} \inenge 0.384 \]
\[ \gamma = \arccos(0.384) \inenge 67.64^\circ \]
Saka, vhekita yeR inobuda ine hurefu hwemayuniti angangoita 7.81 uye mafambiro ayo anoenderana ne x, y, uye z axes ari 59.50°, 39.50°, uye 67.64°.
Mubvunzo 3
Zvichipiwa mavector maviri:
– P ine hukuru hwemayuniti mana uye inoumba kona ye45° kune x-axis yakanaka.
– Q ine hukuru hwemayuniti matanhatu uye inoumba kona ye120° kune x-axis yakanaka.
Sarudza vhekitari inobuda R.
Kukurukurirana
Kutanga, tinoparadzanisa vector kuita zvikamu zvayo zve x na y:
– Vekitori P : \(P_x = 4\cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\), \(P_y = 4\sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\).
– Vekitori Q : \(Q_x = 6\cos(120^\circ) = 6 \cdot \left(-\frac{1}{2}\right) = -3\), \(Q_y = 6\sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} \approx 5.2\).
Heino huwandu hwezvikamu:
\[ R_x = P_x + Q_x = 2.83 – 3 = -0.17 \]
\[ R_y = P_y + Q_y = 2.83 + 5.2 = 8.03 \]
Zvadaro, vhekitari inoguma R ndeiyi:
\[ R = (-0.17, 8.03) \]
Kuti uverenge urefu (modulus) yevector R:
\[ |R| = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.0289 + 64.48} = \sqrt{64.509} \inenge 8.03 \]
Kutungamirirwa kwevector R:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{8.03}{-0.17} = -47.24 \]
\[ \theta = \arctan(-47.24) \approx -88.99^\circ \]
Zvisinei, kona iyi inoyerwa nezve x-axis isina kunaka, saka kona chaiyo muchirevo chedambudziko ndeiyi:
\[ 180^\circ – 88.99^\circ \approx 91.01^\circ \]
Saka, vhekita R inobuda ine urefu hwemayuniti angangoita 8.03 uye inoumba kona ye91.01° ine x-axis yakanaka.
Chinyorwa chino chakurukura nezvekuwedzera kwevector maererano nezvikamu, zvichipa mienzaniso yakawanda yezvinetso nemhinduro. Nzira iyi maererano nezvikamu inobatsira zvikuru mukurerutsa kuverenga uye kupa nzira yakarongeka yekugadzirisa matambudziko evector muchikamu chemasvomhu chenzvimbo.
