Muenzaniso weMatambudziko eKukurukura nezveZvinzvimbo
Mavector ipfungwa huru mumasvomhu nefizikisi, inomiririra huwandu hwakananga uye hukuru. Mukushandiswa kwakasiyana-siyana, mavector anowanzo shandiswa kutsanangura nzvimbo, kumhanya, simba, nezvimwe zvakawanda. Pakati pemhando dzakasiyana dzemavector, mavector enzvimbo anoita basa rakakosha mukuona nzvimbo yenzvimbo munzvimbo.
Tsanangudzo yeChinzvimbo Vector
Vector yenzvimbo ivhector inotsanangura nzvimbo yepoindi kana tichienzanisa nekwakabva muhurongwa hwemakodheni. Kazhinji, vector yenzvimbo inonyorwa muchimiro cheCartesian coordinate seizvi:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
Pano, \(\mathbf{r}\) ndiyo position vector, \(x\), \(y\), uye \(z\) ndiwo macomponents ayo ari padivi pe \(x\), \(y\), uye \(z\) axes, zvichiteerana, nepo \(\mathbf{i}\), \(\mathbf{j}\), uye \(\mathbf{k}\) ari ma unit vectors ari parallel to the coordinate axes, zvichiteerana. Munzvimbo ine mativi maviri, \(z\) chikamu hachiwanikwe, saka position vector inova:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} \]
Zvikumbiro zveVector yeChinzvimbo
Semuenzaniso, mufizikisi, mavekitari enzvimbo anoita basa rakakosha mukutsanangura kufamba kwezvinhu. Nzvimbo yechinhu maererano nekwakabva (nzvimbo yekunongedzera) inogona kumirirwa nevekitari yenzvimbo. Uyezve, muinjiniya yemakanika, kuverenga kwemasimba nenguva kunowanzo sanganisira kushandiswa kwemavekitari enzvimbo.
Mibvunzo yeMienzaniso neKukurukurirana kweZvinzvimbo
Mubvunzo 1
Ngatitii pane mapoinzi maviri munzvimbo ye3D, poindi A ine macoordinates \( (1, 2, 3) \) uye poindi B ine macoordinates \( (4, 0, -2) \). Sarudza mavector enzvimbo dzemapoinzi A naB. Pamusoro pezvo, verenga poindi yekubatanidza A nepoindi B.
Kukurukurirana:
Vekitari yenzvimbo yepoindi A:
\[ \mathbf{r_A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \]
Vekitari yenzvimbo yepoindi B:
\[ \mathbf{r_B} = 4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k} \]
Tevere, kuti tiwane vector inobatanidza poindi A kuenda ku poindi B (inonzi \(\mathbf{AB}\)), tinofanira kubvisa position vector yaA kubva kuposition vector yaB:
\[ \mathbf{AB} = \mathbf{r_B} – \mathbf{r_A} \]
Saka, kutsiva maveji maviri enzvimbo ari pamusoro:
\[ \mathbf{AB} = (4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k}) – (1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \]
\[ \mathbf{AB} = (4 – 1)\mathbf{i} + (0 – 2)\mathbf{j} + (-2 – 3)\mathbf{k} \]
\[ \mathbf{AB} = 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \]
Saka, nzvimbo yekubatanidza vector A kuenda kuB ndeye \( 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \).
Mubvunzo 2
Kana poindi P iri pa \((2, 3)\) mu 2D plane, tsvaga kureba (norm) kwenzvimbo yevector \(\mathbf{r_P}\).
Kukurukurirana:
Chinzvimbo vekitori yepoindi P:
\[ \mathbf{r_P} = 2\mathbf{i} + 3\mathbf{j} \]
Kureba kwenzvimbo yevector \(\mathbf{r_P}\) kunogona kuverengerwa uchishandisa fomura yevector norm (kana urefu):
\[ \| \mathbf{r_P} \| = \sqrt{x^2 + y^2} \]
Tsiva kukosha kwe \(x\) uye \(y\):
\[ \| \mathbf{r_P} \| = \sqrt{2^2 + 3^2} \]
\[ \| \mathbf{r_P} \| = \sqrt{4 + 9} \]
\[ \| \mathbf{r_P} \| = \sqrt{13} \]
Saka, kureba kwenzvimbo yevector \(\mathbf{r_P}\) ndi \(\sqrt{13}\).
Mubvunzo 3
Ngatitii poindi Q iri pa \( (5, -4, 2) \). Tsvaga kona iri pakati pevector yenzvimbo \(\mathbf{r_Q}\) uye \(x\) axis.
Kukurukurirana:
Chinzvimbo vekitori yepoindi Q:
\[ \mathbf{r_Q} = 5\mathbf{i} – 4\mathbf{j} + 2\mathbf{k} \]
Kuti tiwane kona iri pakati pevector \(\mathbf{r_Q}\) ne \(x\) axis, tinogona kushandisa pfungwa yechigadzirwa chedot. Kutanga, tinosarudza chigadzirwa chedot pakati pe \(\mathbf{r_Q}\) uye \(\mathbf{i}\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5\mathbf{i} \cdot \mathbf{i} + (-4\mathbf{j} \cdot \mathbf{i}) + 2\mathbf{k} \cdot \mathbf{i}]
Sezvo \(\mathbf{i} \cdot \mathbf{i} = 1\), \(\mathbf{j} \cdot \mathbf{i} = 0\), uye \(\mathbf{k} \cdot \mathbf{i} = 0\), saka:
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5 \]
Maitiro e \(\mathbf{r_Q}\):
\[ \| \mathbf{r_Q} \| = \sqrt{5^2 + (-4)^2 + 2^2} \]
\[ \| \mathbf{r_Q} \| = \sqrt{25 + 16 + 4} \]
\[ \| \mathbf{r_Q} \| = \sqrt{45} \]
\[ \| \mathbf{r_Q} \| = 3\sqrt{5} \]
Kazhinji kacho \(\mathbf{i}\) i1, nekuti \(\mathbf{i}\) iyuniti yevekita.
Kushandisa fomura yechigadzirwa chedoti kuti uwane kona \(\theta\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = \| \mathbf{r_Q} \| \| \mathbf{i} \| \cos\theta\]
\[ 5 = 3\sqrt{5} \cos\theta \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \]
\[ \cos\theta = \frac{5\sqrt{5}}{15} \]
\[ \cos\theta = \frac{\sqrt{5}}{3} \]
Saka kona \(\theta\) iri pakati penzvimbo yevector \(\mathbf{r_Q}\) uye \(x\) axis ndeiyi:
\[ \theta = \cos^{-1} \left(\frac{\sqrt{5}}{3}\right) \]
Mhedziso
Mavector enzvimbo anoita basa rakakosha musainzi neinjiniya, kunyanya pakuona nzvimbo yezvinhu munzvimbo inotevedzana. Mienzaniso iri pamusoro inoratidza maverengero enzvimbo, kureba kwadzo, uye makona ari pakati padzo neaxes dzinoenderana. Kunzwisisa pfungwa idzi dzakakosha kunokosha mukugadzirisa matambudziko akasiyana-siyana anosanganisira nzvimbo nemacoordinates mumasvomhu nefizikisi.
