Mibvunzo Yemuenzaniso Kukurukura Mashoko, Zvinyorwa, uye Mhando dzeVectors
Kunyora mavector uye kunzwisisa kwawo zvakakosha mumapazi akasiyana esainzi, kunyanya fizikisi nemasvomhu. Kushandisa mavector nemazvo kunogona kubatsira mukuongorora matambudziko nekuwana mhinduro dzinoshanda. Chinyorwa chino chinokurukura mazwi nekunyora zvine chekuita nemavector, zvichiratidza mienzaniso netsananguro dzakakwana.
Mazwi eVector
Kuti tinzwisise mavector, tinofanira kutanga tanzwisisa mazwi ekutanga:
1. Vector: Huwandu hune hukuru (hukuru hukuru) uye gwara. Vector dzinowanzo fananidzirwa nemabhii matema akadai saA, a, kana nechiratidzo chemuseve chiri pamusoro pawo senge \(\vec{A}\).
2. Hukuru (Hukuru Hwakawanda): Uku ndiko kureba kana saizi yevector. Inoratidzwa ne | A | kana \(\|\vec{A}\|\).
3. Musoro neMuswe: Mumifananidzo, mavector anoratidzwa semiseve. Panotangira museve panonzi muswe uye panogumira museve panonzi musoro.
4. Mavekitari Akafanana: Mavekitari akaenzana kana kuti ari pamutsara mumwe chete wekuita.
5. Mavhekitari eCollinear: Mavhekitari ari pamutsetse mumwe wakatwasuka.
6. Resultant Vector: Vector imwe chete ine mhedzisiro yakafanana nemhedzisiro yakabatanidzwa yevector mbiri kana kupfuura.
Kunyora kweVector
Kunyora mavhekitari kune mitemo yakati wandei inofanirwa kunzwisiswa kuti idudzire uye inyore mavhekitari nemazvo.
1. Mabhii Matema neMuseve: Mavhekitori anowanzo ratidzwa nemabhii matema kana miseve. Mienzaniso: A , B , kana \(\vec{A}\).
2. Makoordinati eVector: Mavector ari munzvimbo ine mativi maviri (2D) anoratidzwa se \(\vec{A} = (A_x, A_y)\), nepo ari munzvimbo ine mativi matatu (3D) anoratidzwa se \(\vec{A} = (A_x, A_y, A_z)\).
3. Mavekitari eBasis: Munzvimbo ye2D ne3D, mavekitari eBasis anoshandiswa kazhinji ndi \(\vec{i}\), \(\vec{j}\), uye \(\vec{k}\), ayo anoreva nzira dze x, y, uye z, zvichiteerana.
4. Mashandiro eVector:
– Kuwedzera: \(\vec{A} + \vec{B}\)
– Kubvisa : \(\vec{A} – \vec{B}\)
– Kuwanda kweScalar: \(k\vec{A}\)
– Kuwanda kweDot (chigadzirwa chedot): \(\vec{A} \cdot \vec{B}\)
– Kuwanda kweMuchinjiko (muchinjiko): \(\vec{A} \times \vec{B}\)
Mhando dzeVector
Mhando dzakasiyana dzemavector dzinogona kuwanikwa zvichienderana nemamiriro ezvinhu uye hunhu hwadzo:
1. Zero Vector: Vector ine hukuru hwe0 uye isina gwara. Inoratidzwa ne0 kana \(\vec{0}\).
2. Chiratidzo cheUnit: Chiratidzo chine hukuru hwe1. Chinowanzo shandiswa kuratidza divi.
3. Chinzvimbo Vector: Vector inoratidza nzvimbo yepoindi kana tichienzanisa nekwakabva (0,0,0).
4. Mavekitari akafanana uye asingaenderane: Mavekitari ari munzira imwe chete uye akasiyana, asi ari munzira imwe chete yekushanda.
5. Maveji eCoplanar: Maveji ari muchikamu chimwe chete.
Mibvunzo yemuenzaniso nekukurukurirana
Mubvunzo 1: Kuverenga Ukuru hweVekitori
Ukuru hwevector \(\vec{A} = (3, 4)\) chii?
Jawaban:
Kuti tiverenge hukuru hwevector \(\vec{A}\), tinoshandisa fomura:
\[\|\vec{A}\| = \sqrt{A_x^2 + A_y^2}\]
Tsiva ma "values" mufomura:
\[\|\vec{A}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
Saka, hukuru hwevector \(\vec{A}\) ndi5.
Mubvunzo 2: Kuwedzera nekubvisa maVectors
Zvichipiwa mavector maviri \(\vec{A} = (2, 3)\) uye \(\vec{B} = (1, -1)\). Verenga \(\vec{A} + \vec{B}\) uye \(\vec{A} – \vec{B}\).
Jawaban:
Kuwedzera mavector \(\vec{A}\) uye \(\vec{B}\):
\[\vec{A} + \vec{B} = (2, 3) + (1, -1) = (2 + 1, 3 – 1) = (3, 2)\]
Kubvisa mavector \(\vec{A}\) uye \(\vec{B}\):
\[\vec{A} – \vec{B} = (2, 3) – (1, -1) = (2 – 1, 3 – (-1)) = (1, 4)\]
Saka, \(\vec{A} + \vec{B} = (3, 2)\) uye \(\vec{A} – \vec{B} = (1, 4)\).
Mubvunzo 3: Chigadzirwa cheDot
Verenga dot product yemavector maviri \(\vec{A} = (2, 3)\) uye \(\vec{B} = (1, 4)\).
Jawaban:
Chigadzirwa che dot che mavector maviri ndeichi:
\[\vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y\]
Kutsiva kukosha:
\[\vec{A} \cdot \vec{B} = 2 \cdot 1 + 3 \cdot 4 = 2 + 12 = 14\]
Saka, dot product ye \(\vec{A}\) uye \(\vec{B}\) i14.
Mubvunzo 4: Chigadzirwa Chinosanganiswa
Tichipa mavector maviri munzvimbo ine mativi matatu \(\vec{A} = (1, 2, 3)\) uye \(\vec{B} = (4, 5, 6)\). Verenga chigadzirwa chakasiyana \(\vec{A} \times \vec{B}\).
Jawaban:
Chigadzirwa che mavector maviri ari munzvimbo ine mativi matatu chinotsanangurwa sechinhu chinotsanangura matrix inotevera:
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\kuguma{vmatrix}
\]
Kune mavector \(\vec{A}\) uye \(\vec{B}\):
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
1 & 2 & 3 \\
4 ne5 & 6
\kuguma{vmatrix}
\]
Yakaverengerwa se:
\[
\vec{A} \nguva \vec{B} = \vec{i}(2 \cdot 6 – 3 \cdot 5) – \vec{j}(1 \cdot 6 – 3 \cdot 4) + \vec{k}(1 \cdot 5 – 2 \cdot 4)
\]
\[
= \vec{i}(12 – 15) – \vec{j}(6 – 12) + \vec{k}(5 – 8)
\]
\[
= \vec{i}(-3) – \vec{j}(-6) + \vec{k}(-3)
\]
\[
= -3\vec{i} + 6\vec{j} – 3\vec{k}
\]
Saka, mubatanidzwa we \(\vec{A}\) uye \(\vec{B}\) ndi \(\vec{A} \times \vec{B} = (-3, 6, -3)\).
Pakugadzirisa matambudziko evector, kunzwisisa pfungwa huru uye mazwi ekutanga ndiyo nzvimbo huru yekutanga. Chinyorwa chino chinangwa ndechekupa vaverengi kunzwisisa mashandiro akasiyana-siyana evector nemhando dzawo dzakasiyana, izvo zvichave zvakakosha mukuongorora kwemasvomhu nemuviri.
