Zvikamu zveVector: Zvinokosha, Tsanangudzo, uye Mashandisirwo
Mavector ipfungwa huru mumasvomhu, fizikisi, uye mainjiniya. Munzvimbo dzakasiyana dzesainzi, anowanzo shandiswa kutsanangura huwandu hune hukuru uye gwara. Muchinyorwa chino, tichaongorora zvikamu zvevector: kutsanangura kuti vector chii, nzira yekuparadzanisa vector muzvikamu zvayo, uye kuongorora mashandisirwo akasiyana-siyana uye zvinorehwa nemavector muhupenyu hwezuva nezuva nesainzi.
Kunzwisisa Mavekta
Vector huwandu husina kukosha chete (hukuru) asiwo gwara. Kusiyana nemascalars, ayo ane kukosha chete (senge tembiricha kana huremu), mavectors ane hunhu hukuru uhwu uye anoshandiswa kumiririra zviitiko apo gwara riri chinhu chakakosha, senge kumhanya, simba, uye kutama.
Pamasvomhu, vhekitari iri munzvimbo ine mativi maviri (2D) inogona kuratidzwa se \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), uko \(v_x\) uye \(v_y\) zviri zvikamu zvevhekitari \(\mathbf{v}\) mu x- uye y-directions. Munzvimbo ine mativi matatu (3D), vhekitari inogona kuratidzwa se \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\).
Kumiririrwa kweVector uye Zvikamu
Kuti tinzwisise pfungwa yezvikamu zvevector, tinofanira kuziva kuti mavector anogona kukamurwa kuita zvikamu zvinoenderana ne coordinate axis yega yega. Semuenzaniso, munzvimbo ine mativi maviri, vector \(\mathbf{v}\) inogona kukamurwa kuita zvikamu zviviri: \(v_x\) (chikamu chiri mu x-direction) uye \(v_y\) (chikamu chiri mu y-direction).
Pakuongorora geometrics, kana tikaronga vector paCartesian coordinate plane, inogona kuenzaniswa nemuseve unonongedza kubva pakutanga \((0,0)\) kusvika papoindi \((v_x, v_y)\). Zvikamu \(v_x\) uye \(v_y\) zvinogona kuonekwa sehurefu hwemapurojekiti evector pa x- uye y-axes.
Munzvimbo ine mativi matatu, vhekitari inogona kukamurwa kuita zvikamu zvitatu: \(v_x\) (chikamu che x-direction), \(v_y\) (chikamu che y-direction), uye \(v_z\) (chikamu che z-direction). Nemamwe mashoko, vhekitari iri munzvimbo ine mativi matatu inogona kumirirwa nemuseve unonongedza kubva pakutanga \((0,0,0)\) kusvika panzvimbo \((v_x, v_y, v_z)\).
Hukuru uye Kutungamirirwa kweVectors
Kuti tiverenge hukuru kana hurefu hwevector \(\mathbf{v}\), tinoshandisa fomura iyi:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
\]
yenzvimbo ine mativi maviri, uye:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
panzvimbo ine mativi matatu. Huwandu hwevector iyi hunowanzo nzi hukuru hwayo uye hunoratidza kuti vector yakareba sei.
Kutungamira kwevector kunogona kuratidzwa maererano nekona yayo maererano nema coordinate axes. Munzvimbo ine mativi maviri, kutungamira kwevector \(\mathbf{v}\) inoumba angle \(\theta\) ine x-axis kunogona kuverengerwa uchishandisa trigonometry:
\[
\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)
\]
Munzvimbo ine mativi matatu, kusarudza kwakananga kwakaoma, nekuti tinofanira kuverenga makona ane coordinate axis yega yega. Kazhinji, system yakatenderera inoshandiswa kuratidza gwara munzvimbo ine mativi matatu.
Mashandiro eVectors
Kuwedzera nekubvisa
Kuwedzera mavector maviri kunoitwa nekuwedzera zvikamu zvemavector ese ari maviri. Semuenzaniso, kana \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \end{bmatrix}\) uye \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), saka:
\[
\mathbf{u} + \mathbf{v} = \kutanga{bmatrix} u_x + v_x \\ u_y + v_y \kuguma{bmatrix}
\]
Kubvisa kwevector kunoverengerwa nenzira yakafanana:
\[
\mathbf{u} – \mathbf{v} = \kutanga{bmatrix} u_x – v_x \\ u_y – v_y \kuguma{bmatrix}
\]
Kuwedzera kweScalar
Kuwanda kwevector ne scalar (nhamba imwe chete) kunoitwa nekuwanda kwechikamu chimwe nechimwe chevector ne scalar. Semuenzaniso, kana \(k\) iri scalar uye \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), saka:
\[
k \cdot \mathbf{v} = \begin{bmatrix} k \cdot v_x \\ k \cdot v_y \end{bmatrix}
\]
Kuwanda kweDot neCross
Munzvimbo ine mativi matatu, kune mhando mbiri dzekuwedzera kwevector: kuwedzera kwedot uye kuwedzera kwecross.
1. Kuwanda kweMadonhwe:
Chigadzirwa che dot chemavector maviri \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix}\) uye \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\) chinotsanangurwa se:
\[
\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z
\]
Mhedzisiro yechigadzirwa chedot iscalar. Chigadzirwa chedot chinowanzo shandiswa kuona kuti mavector maviri akafanana kana kuti akafanana zvakadii.
2. Kuwanda kwezvikamu:
Chigadzirwa che mavector maviri ari munzvimbo ine mativi matatu chinoburitsa vector itsva yakatarisana nemavector ese ekutanga. Kana \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix}\) uye \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\), saka chigadzirwa che cross chinotsanangurwa se:
\[
\mathbf{u} \nguva \mathbf{v} = \kutanga{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_x & u_y & u_z \\
v_x & v_y & v_z
\kuguma{vmatrix}
\]
Kugadziriswa kweVector
Normalization inzira yekushandura vector kuita unit vector (vector yehurefu 1) ine divi rimwe chete. Vector ye unit \(\mathbf{\hat{v}}\) ye \(\mathbf{v}\) inowanikwa nekukamura chimwe nechimwe chezvikamu zvayo nehurefu (hukuru) hwevector:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}
\]
Mashandisirwo eVectors muhupenyu hwezuva nezuva nesainzi
Mavector ane mashandisirwo akasiyana-siyana muhupenyu hwezuva nezuva nesainzi. Heano mimwe mienzaniso:
1. Fizikisi:
Mufizikisi, mavector anoshandiswa kutsanangura huwandu hwakasiyana-siyana hwakadai sekukurumidza, kukurumidza, simba, uye kumhanya. Semuenzaniso, kufamba kwechinhu kunogona kuongororwa uchishandisa mavector ekumhanya uye kukurumidza.
2. Maitiro:
Muinjiniya, mavector anoshandiswa pakuongorora chimiro, kugadzira michina, uye mamwe mashandisirwo akasiyana-siyana einjiniya. Semuenzaniso, kuongorora kushushikana uye kusvuta muchinhu kunowanzo sanganisira kushandiswa kwemavector.
3. Mifananidzo yeKombuta:
Mavector anoshandiswawo mumakombiyuta kutsanangura nzvimbo, kurongeka, uye kufamba kwezvinhu. Mukugadzira mifananidzo, mavector anoshandiswa pakushandura zvinhu zvakaita sekushandura, kutenderera, uye kukura.
4. Kufamba:
Mavector anoshandiswa mumasystem ekufamba-famba kuona divi uye daro riri pakati penzvimbo mbiri. GPS nedzimwe nzira dzekufamba-famba dzinoshandisa mavector kuverenga nzira nekutungamira vashandisi.
5. Hupfumi:
Muzvehupfumi, mavector anogona kushandiswa kutsanangura zvinodiwa nevatengi kana maportfolio ekudyara. Kuongororwa kwedata rinoshanduka-shanduka kunowanzo sanganisira kushandiswa kwemavector.
Mhedziso
Mavector ipfungwa inokosha uye inoshanda zvakasiyana-siyana mumasvomhu nedzimwe nzvimbo dzakawanda dzesainzi. Nekunzwisisa zvikamu zvevectors uye mashandiro akasiyana-siyana anogona kuitwa pavari, tinogona kushandisa pfungwa iyi kugadzirisa matambudziko akasiyana-siyana anoshanda uye esainzi. Nekumiririra kwavo kwakasimba kwemasvomhu, mavectors anopa chishandiso chinoshanda chekutsanangura nekuongorora zviitiko zvakasiyana-siyana zvinosanganisira huwandu nemirairo.
