Mavector eChinzvimbo: Zvinokosha, Mashandisirwo, uye Mienzaniso muhupenyu hwezuva nezuva
Pendauluan
Chiratidzo chenzvimbo (position vector) ipfungwa inokosha mumasvomhu nefizikisi inoshandiswa kutsanangura nzvimbo yenzvimbo munzvimbo. Zvichitaurwa zviri nyore, chiratidzo chenzvimbo chinogona kufungwa nezvacho semuseve unonongedza kubva pakutanga (kazhinji kwakabva) kusvika panzvimbo yakatarwa. Chinyorwa chino chichakurukura tsananguro yechiratidzo chenzvimbo, zvikamu zvacho, maitiro ekuchiverenga, uye mashandisirwo azvo muhupenyu hwezuva nezuva.
Tsanangudzo yeChinzvimbo Vector
Vector yenzvimbo ivhector inobatanidza mavambo kune poindi iri muchadenga. Pakushanda munzvimbo ine mativi maviri (2D), vector yenzvimbo inoratidzwa sepeya yakarongwa \((x, y)\), apo \(x\) uye \(y\) zviri macoordinates epoindi. Munzvimbo ine mativi matatu (3D), vector yenzvimbo inoratidzwa setriplet yakarongwa \((x, y, z)\).
Semuenzaniso, kana tine poindi A pa coordinates (3, 4) munzvimbo ye2D, saka position vector inobatanidza origin (0,0) kune poindi A i \(\mathbf{r} = 3\mathbf{i} + 4\mathbf{j}\), apo \(\mathbf{i}\) uye \(\mathbf{j}\) ari ma unit vectors ari pa axes dze \(x\) uye \(y\) .
Zvikamu zveVector zveChinzvimbo
Vekitari yenzvimbo ine zvikamu zvinomiririra madaro ari pamwe chete nemaaxes e coordinate. Munzvimbo ye2D, vekitari yenzvimbo \(\mathbf{r}\) inogona kuratidzwa seizvi:
\[
\mathbf{r} = x\mathbf{i} + y\mathbf{j}
\]
Pano, \(x\) ndicho chikamu chevector yenzvimbo ichitevedza axis ye\(x\), uye \(y\) ndicho chikamu chevector yenzvimbo ichitevedza axis ye\(y\).
Munzvimbo ye3D, vhekitari yenzvimbo \(\mathbf{r}\) inoratidzwa seizvi:
\[
\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}
\]
Pano, \(x\), \(y\), uye \(z\) ndizvo zvikamu zvevector yenzvimbo pamwe chete neaxes dze \(x\), \(y\), uye \(z\) zvichiteerana, nepo \(\mathbf{k}\) iri vector yeyuniti pamwe chete neaxes dze \(z\).
Maitiro Ekuverenga Nzvimbo Vector
Kuverenga tarisiro yenzvimbo kunosanganisira kuona daro kubva pakutanga kusvika panzvimbo iri kutaurwa muCartesian coordinates. Semuenzaniso, kana poindi B iri pamakoniti (5, 7) munzvimbo ye2D, tarisiro yenzvimbo inobatanidza mavambo (0, 0) nepoindi B ndeiyi:
\[
\mathbf{r_B} = 5\mathbf{i} + 7\mathbf{j}
\]
Kuti tiverenge kureba kwenzvimbo yevector (kana hukuru), tinoshandisa dzidziso yePythagorean. Kureba kwenzvimbo yevector \(\mathbf{r}\) munzvimbo ye2D kunopiwa na:
\[
|\mathbf{r}| = \sqrt{x^2 + y^2}
\]
Munzvimbo ye3D, kureba kwenzvimbo yevector \(\mathbf{r}\) kunoverengerwa seizvi:
\[
|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}
\]
Semuenzaniso, kana paine poindi C pa coordinates (3, 4, 5) munzvimbo ye3D, kureba kwenzvimbo yevector \(\mathbf{r_C}\) ndekwekuti:
\[
|\mathbf{r_C}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07
\]
Mashandisirwo eZvinhu zveMaitiro muhupenyu hwezuva nezuva
Mavector enzvimbo ane mashandisirwo akawanda anoshanda muzvikamu zvakasiyana-siyana. Heano mimwe mienzaniso yekushandiswa kwawo muhupenyu hwezuva nezuva:
1. Kufamba uye GPS
Mumasystem ekufamba-famba akadai seGPS, mavector enzvimbo anoshandiswa kuona nzvimbo yemushandisi kana tichienzanisa nemasatellite eGPS. Ruzivo urwu rwenzvimbo runoshandiswa kuverenga daro uye gwara rekufamba.
2. Uinjiniya hwezvivakwa nekuvaka
Mainjiniya ekuvaka nevanovaka dzimba vanoshandisa mavector enzvimbo kugadzira zvivakwa nezvivakwa. Mavector aya anobatsira kuona nzvimbo dzakasiyana dzezvivakwa.
3. Ruzivo rwenyeredzi
Mukuongorora nyeredzi, ma "position vectors" anoshandiswa kutsanangura nzvimbo dzenyeredzi, mapuraneti, nezvimwe zvinhu zvekudenga zvichienzaniswa nePasi kana pakati penyika yezuva.
4. Zvenyama
Mufizikisi, maveki enzvimbo akakosha pakuongorora kufamba. Semuenzaniso, pakuongorora kufamba kwechinhu, maveki enzvimbo anoshandiswa kuona nzvimbo yechinhu panguva dzakasiyana.
5. Marobhoti
Mumarobhoti, mavector enzvimbo anoshandiswa kudzora kufamba kwerobhoti. Mavector aya anobatsira robhoti kuziva nzvimbo chaiyo uye gwara rekufamba kwaro.
Mibvunzo yemuenzaniso nemhinduro
Mubvunzo unotevera muenzaniso wekujekesa kunzwisisa kwemavekitari enzvimbo.
Mubvunzo:
Poindi P iri pamakoneti (2, 3) uye poindi Q iri pamakoneti (5, 7) munzvimbo ye2D. Sarudza nzvimbo yevector inobatanidza poindi P nepoindi Q uye verenga kureba kwevector.
Mhinduro:
Veki yenzvimbo \(\mathbf{PQ}\) ndiyo veki inobatanidza P nepoindi Q. Tinogona kuverenga zvikamu zveveki \(\mathbf{PQ}\) nekubvisa macoordinates eP kubva kumacoordinates eQ:
\[
\mathbf{PQ} = (5 – 2)\mathbf{i} + (7 – 3)\mathbf{j} = 3\mathbf{i} + 4\mathbf{j}
\]
Kureba kwevector yenzvimbo \(\mathbf{PQ}\) ndekwekuti:
\[
|\mathbf{PQ}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Saka, nzvimbo yekubatanidza vhekitari P kuenda kupoindi Q ndeye \(3\mathbf{i} + 4\mathbf{j}\) uye kureba kwevhekitari mayuniti mashanu.
Mhedziso
Chiratidzo chenzvimbo ipfungwa huru inoshandiswa kutsanangura nzvimbo yenzvimbo munzvimbo. Kunzwisisa zvikamu uye maverengerwo echitarisiko chenzvimbo kwakakosha kune akasiyana mashandisirwo anoshanda, kubva pakufamba kusvika pakuongorora kwemuviri. Kunzwisisa izvi zvakakosha kunoita kuti zvive nyore kunzwisisa nekushandisa pfungwa yemagadzirirwo enzvimbo muhupenyu hwezuva nezuva uye mumabasa ehunyanzvi.
