Muenzaniso weMubvunzo weKukurukurirana pamusoro peVector Addition
Pendauluan
Mavector ipfungwa huru mumasvomhu nefizikisi, anowanzo shandiswa kumiririra huwandu hune hukuru negwara, zvakaita sekumhanya, simba, uye kutama. Muzviitiko zvakawanda, tinowanzo sangana nemamiriro ezvinhu patinoda kuwedzera mavector maviri kana kupfuura. Chinyorwa chino chichakurukura mienzaniso yakati wandei yezvinetso zvekuwedzera mavector nemhinduro dzawo kuti tinzwisise zvakadzama pfungwa iyi.
Kunzwisisa Kuwedzera Vector
Mumasvomhu, kuwedzera kwevector kunogona kuitwa uchishandisa nzira mbiri huru: nzira yetriangle uye nzira yeparallelogram. Imwe nzira inowanzoshandiswa inzira ye component. Heino tsananguro pfupi yenzira idzi nhatu:
1. Nzira yeTriangle: Munzira iyi, magumo evector yekutanga anoiswa panzvimbo yekutanga yevector yechipiri. Mhedzisiro yekuwedzera ivector inobatanidza nzvimbo yekutanga yevector yekutanga nemagumo evector yechipiri.
2. Nzira yeParalelogram: Mavector ese ari maviri akaiswa panzvimbo imwe chete yekutanga. Mugumisiro wekuwedzera ivector ye diagonal yeparalelogram inoumbwa nemavector maviri.
3. Nzira yeComponent: Vector inokamurwa kuita zvikamu zvichitevedza x na y axes. Zvikamu izvi zvinosanganiswa pamwe chete, uye huwandu hwezvikamu hunoshandiswa kuona vector inobuda.
Mibvunzo yemuenzaniso nekukurukurirana
Zvino, ngatikurukurei mimwe mienzaniso yezvinetso zvekuwedzera vector tichishandisa nzira nhatu dziri pamusoro.
Mubvunzo 1: Kuwedzera Vector Uchishandisa Nzira yeTriangle
Mubvunzo:
Zvichipiwa mavector maviri A naB apo A = 5i + 3j naB = -2i + 4j. Sarudza huwandu hwemavector A + B.
Kukurukurirana:
Nzira yetriangle inosimbisa kubatana kwevector zvakananga, asi kana iri mavector akavakirwa pazvikamu, tinogona kubatanidza chikamu chimwe nechimwe zvakananga.
1. Zvikamu zve x zveA naB:
\( A_x = 5, B_x = -2 \)
Saka, \( A_x + B_x = 5 – 2 = 3 \)
2. Zvikamu zve y zveA naB:
\( A_y = 3, B_y = 4 \)
Saka, \( A_y + B_y = 3 + 4 = 7 \)
Saka, mhedzisiro yekuwedzera mavectors A + B ndeiyi:
\[
A + B = 3i + 7j
\]
Mubvunzo 2: Kuwedzera Vector Uchishandisa Nzira yeParallelogram
Mubvunzo:
Zvichipiwa mavector maviri, C = 4i + j uye D = 2i + 5j. Tsvaga huwandu hwemavector C + D uchishandisa nzira yeparallelogram.
Kukurukurirana:
Nekushandisa nzira yeparallelogram, mavector ese ari maviri anoiswa panzvimbo imwe chete yekutanga, asi huwandu hwezvikamu hunoramba hwakafanana nehwekushandisa nzira yetriangle muCartesian coordinates.
1. Zvikamu zve x zve C na D:
\( C_x = 4, D_x = 2 \)
Saka, \( C_x + D_x = 4 + 2 = 6 \)
2. Zvikamu zve y zve C na D:
\( C_y = 1, D_y = 5 \)
Saka, \( C_y + D_y = 1 + 5 = 6 \)
Saka, mhedzisiro yekuwedzera mavector C + D ndeiyi:
\[
C + D = 6i + 6j
\]
Mubvunzo 3: Kuwedzera Vector Uchishandisa Nzira yeComponent
Mubvunzo:
Zvichipiwa mavector maviri E = 7i – 2j uye F = -3i + 6j. Tsvaga huwandu hwemavector E + F uchishandisa nzira yechikamu.
Kukurukurirana:
Nzira yezvikamu inoda mapfupiso akasiyana echikamu chimwe nechimwe.
1. Zvikamu zve x zveE naF:
\( E_x = 7, F_x = -3 \)
Saka, \( E_x + F_x = 7 – 3 = 4 \)
2. Zvikamu zve y zveE naF:
\( E_y = -2, F_y = 6 \)
Saka, \( E_y + F_y = -2 + 6 = 4 \)
Saka, mhedzisiro yekuwedzera mavectors E + F ndeiyi:
\[
E + F = 4i + 4j
\]
Mubvunzo 4: Kuwedzera kweMavector Asiri eCartesian
Mubvunzo:
Zvichipiwa mavector maviri G naH ane hukuru negwara sezvinotevera: G ine hukuru hwemayunitsi mashanu negwara remadhigirii makumi matatu, nepo H ine hukuru hwemayunitsi gumi negwara remadhigirii zana nemakumi maviri. Sarudza huwandu hwevector hweG + H.
Kukurukurirana:
Muchiitiko ichi, zvinotanga zvave kudikanwa kushandura vector kuita zvikamu zvayo zve x ne y:
1. Zvikamu zvevector G:
\[
G_x = 5 \cos(30^{\circ}) = 5 \cdot \frac{\sqrt{3}}{2} = 2.5\sqrt{3} \anenge 4.33
\]
\[
G_y = 5 \sin(30^{\circ}) = 5 \cdot \frac{1}{2} = 2.5
\]
2. Zvikamu zvevector H:
\[
H_x = 10 \cos(120^{\circ}) = 10 \cdot (-0.5) = -5
\]
\[
H_y = 10 \sin(120^{\circ}) = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66
\]
Wobva wawedzera zvikamu zve x na y:
Zvikamu zvese zve x:
\[
G_x + H_x = 4.33 – 5 = -0.67
\]
Chikamu chose che y:
\[
G_y + H_y = 2.5 + 8.66 = 11.16
\]
Mhedzisiro yekuwedzera muchimiro cheCartesian vector ndeiyi:
\[
G + H = -0.67i + 11.16j
\]
Kuti uwane hukuru uye gwara rehuwandu, shanduko inoitwa zvakare:
\[
|G + H| = \sqrt{(-0.67)^2 + (11.16)^2} \inenge 11.18
\]
Mirayiridzo ndeiyi:
\[
\theta = \tan^{-1}\left(\frac{11.16}{-0.67}\right) \approx -3.44^\circ + 180^\circ = 176.56^\circ
\]
Saka, mhedzisiro yekuwedzera kwevector G + H inenge iri:
\[
11.18 \, \text{unit} \, \text{negwara} \, 176.56^\circ
\]
Mhedziso
Kuwedzerwa kwevector ipfungwa yakakosha muzvikamu zvakasiyana zvesainzi neinjiniya. Tichishandisa nzira dzetriangles, parallelograms, uye zvikamu, tinogona kunzwisisa nekugadzirisa matambudziko akasiyana-siyana ane chekuita nekuwedzera kwevector. Mumienzaniso iri pamusoro, taona kuti nzira yezvikamu inogona sei kurerutsa zvakanyanya maitiro ekuwedzera kwevector muCartesian coordinates. Tinovimba, nekunzwisisa kwakadzama kwepfungwa idzi, vaverengi vachawana zviri nyore kushandisa nzira dzekuwedzera vector mumamiriro ezvinhu akaoma uye chaiwo.
