Fomura yeVector Inobuda: Matambudziko epfungwa, nzira, uye muenzaniso
Vector huwandu hune hukuru negwara. Mufizikisi nemasvomhu, mavector anowanzo shandiswa kutsanangura zviitiko zvakasiyana-siyana zvakaita sekumhanya, simba, uye kutama. Kuverenga vector inobuda, huwandu hwemavector maviri kana anopfuura, hunyanzvi hwakakosha hunoshandiswa kakawanda mumashandisirwo akasiyana esainzi nehunyanzvi. Chinyorwa chino chichakurukura pfungwa huru yemavector, nzira dzekuverenga vector inobuda, uye kupa mienzaniso yakati wandei yezvinetso kuti kujekeswe kunzwisisa.
Kunzwisisa Mavector uye Mavector Anobuda
vector
Vector chinhu chemasvomhu chine hunhu huviri hukuru:
1. Hukuru: Hukuru hwekukosha kwevector.
2. Kutungamira: Kutungamira kwevector kunoratidza kuti vector iri muchadenga.
Mavector anowanzo ratidzwa semiseve, uko kureba kwemuseve kunomiririra hukuru uye divi remuseve richiratidza divi revector.
Vekitari Inobatsira
Vector inobuda ivector imwe chete inomiririra musanganiswa wemavector maviri kana anopfuura. Maitiro ekuwedzera mavector anozivikanwawo se "vector addition." Kune nzira dzakasiyana siyana dzinogona kushandiswa kuverenga vector inobuda, kusanganisira nzira dzemifananidzo nedzekuongorora.
Nzira Yekuverenga Zvinokonzerwa neVector
Nzira Yemifananidzo
Nzira yemifananidzo inosanganisira kumiririra mavectors geometrically uye kushandisa mitemo yekuwedzera vector kuti uwane mhedzisiro. Mitemo mikuru miviri yenzira yemifananidzo ndeiyi:
1. Nzira yeTriangle: Munzira iyi, vector yechipiri inotorwa kubva kumagumo evector yekutanga. Vector inobuda ivector inotorwa kubva panotangira vector yekutanga kusvika kumagumo evector yechipiri.
2. Nzira yePolygon: Iyi nzira inoshandiswa kuwedzera mavector anopfuura maviri. Mavector anotorwa akatevedzana kubva kumagumo kusvika kumagumo, uye vector inobuda ndiyo vector inobatanidza panotangira vector yekutanga kusvika kumagumo evector yekupedzisira.
Nzira Yekuongorora
Nzira yekuongorora inosanganisira kushandisa masvomhu netrigonometry kuverenga vhekitori inobuda. Nzira mbiri huru dziri munzira yekuongorora ndeidzi:
1. Nzira yeComponent: Munzira iyi, vector yega yega inopatsanurwa kuita zvikamu zvayo pamwe chete ne x- ne y-axes. Zvikamu izvi zvinozowedzerwa pamwe chete kuti pave nezvikamu zvevector inobuda. Pakupedzisira, vector inobuda inoverengwa uchishandisa Pythagorean theorem ne trigonometry.
2. Nzira yeCosine: Iyi nzira inoshandiswa kana hukuru hwemavector maviri nekona iri pakati pawo zvichizivikanwa. Fomura yecosine inoshandiswa kuverenga hukuru hwevector inobuda.
Mafomura eVector Anobudirira
Nzira yeChikamu
Kune mavector maviri \(\mathbf{A}\) uye \(\mathbf{B}\) ane zvikamu:
\[
\mathbf{A} = A_x \heti{i} + A_y \hat{j}
\]
\[
\mathbf{B} = B_x \heti{i} + B_y \hat{j}
\]
Vekitari inobuda \(\mathbf{R}\) ndeiyi:
\[
\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}
\]
Hukuru hwevector inobuda \(\mathbf{R}\) hunogona kuverengerwa uchishandisa dzidziso yePythagorean:
\[
|\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2}
\]
Kutungamira kwevector inobuda kunoonekwa nekona \(\theta\) yakagadzirwa ne x-axis:
\[
\theta = \tan^{-1}\left(\frac{A_y + B_y}{A_x + B_x}\right)
\]
Nzira yeCosine
Kana mavector maviri \(\mathbf{A}\) uye \(\mathbf{B}\) aine masturing \(A\) uye \(B\) uye angle \(\theta\) pakati pawo, masturing evector inobuda \(\mathbf{R}\) ndeekuti:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
Kutungamira kwevector inobuda kunogona kuverengerwa uchishandisa fomura yetrigonometric:
\[
\tan \alpha = \frac{B \sin \theta}{A + B \cos \theta}
\]
Apo \(\alpha\) iri kona inoumbwa nevector inobuda nevector \(\mathbf{A}\).
Muenzaniso weDambudziko reVector Rinobuda
Muenzaniso Mubvunzo 1: Nzira yeComponent
Mubvunzo:
Maveki maviri \(\mathbf{A}\) uye \(\mathbf{B}\) ane zvikamu zvinotevera:
\[
\mathbf{A} = 3\hat{i} + 4\hat{j}
\]
\[
\mathbf{B} = 1\heti{i} + 2\hat{j}
\]
Verenga vhekitari inobuda \(\mathbf{R}\).
Mhinduro:
1. Wedzera zvikamu zviri pa x na y axes:
\[
R_x = A_x + B_x = 3 + 1 = 4
\]
\[
R_y = A_y + B_y = 4 + 2 = 6
\]
2. Verengai hukuru hwevhekitari inobuda:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 7,21
\]
3. Verenga divi revhekitari inobuda:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{6}{4}\right) = \tan^{-1}(1,5) = 56,31^\circ
\]
Saka, vhekitari inobuda \(\mathbf{R}\) ine hukuru hwe7,21 uye divi rayo riri pamadhigirii 56,31 kuenda ku x-axis.
Muenzaniso Mubvunzo 2: Nzira yeCosine
Mubvunzo:
Maveki maviri \(\mathbf{A}\) uye \(\mathbf{B}\) ane magnitudes \(A = 5\) mayuniti, \(B = 7\) mayuniti, uye kona iri pakati pawo i60°. Verenga magnitudes evekitori inobuda \(\mathbf{R}\).
Mhinduro:
1. Shandisa fomura yecosine kuverenga hukuru hwevector inobuda:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
\[
|\mathbf{R}| = \sqrt{5^2 + 7^2 + 2 \cdot 5 \cdot 7 \cdot \cos 60^\circ}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 70 \cdot 0,5}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 35}
\]
\[
|\mathbf{R}| = \sqrt{109} = 10,44 \, \zvinyorwa{chikamu}
\]
Saka, hukuru hwevector inobuda \(\mathbf{R}\) i10,44 units.
Muenzaniso 3: Mhedzisiro yeMavekita Matatu
Mubvunzo:
Maveki matatu \(\mathbf{A}\), \(\mathbf{B}\), uye \(\mathbf{C}\) ane zvikamu zvinotevera:
\[
\mathbf{A} = 2\hat{i} + 3\hat{j}
\]
\[
\mathbf{B} = -1\hat{i} + 4\hat{j}
\]
\[
\mathbf{C} = 3\hat{i} - 2\hat{j}
\]
Verenga vhekitari inobuda \(\mathbf{R}\).
Mhinduro:
1. Wedzera zvikamu zviri pa x na y axes:
\[
R_x = A_x + B_x + C_x = 2 – 1 + 3 = 4
\]
\[
R_y = A_y + B_y + C_y = 3 + 4 – 2 = 5
\]
2. Verengai hukuru hwevhekitari inobuda:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} = 6,4
\]
3. Verenga divi revhekitari inobuda:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{5}{4}\right) = \tan^{-1}(1,25) = 51,34^\circ
\]
Saka, vhekitori inobuda \(\mathbf{
R}\) ine hukuru hwe6,4 uye divi rayo riri pamadhigirii 51,34 kuenda ku x-axis.
Mhedziso
Kuverenga mhedzisiro yevector inyanzvi yakakosha mufizikisi nemasvomhu. Tichishandisa nzira dzemifananidzo kana dzekuongorora, tinogona kuona mhedzisiro yevector mbiri kana kupfuura. Nzira yezvikamu uye nzira yecosine inzira mbiri dzakakosha mukuverenga kwekuongorora dzinotibvumira kuverenga nemazvo hukuru negwara revector yemhedzisiro. Mienzaniso iri pamusoro inoratidza mashandisirwo anoitwa pfungwa idzi, zvichitibatsira kunzwisisa nekushandisa vector mumamiriro akasiyana-siyana esainzi nehunyanzvi.
