Muenzaniso weMubvunzo weKukurukurirana paChikamu cheVector cheVector
Pendauluan
Mumasvomhu nefizikisi, mavector zvinhu zvakakosha zvinomiririra hukuru negwara. Mavector anowanzo shandiswa kutsanangura zviitiko zvakasiyana-siyana zvakaita sekumhanya, simba, uye kutama munzvimbo ine mativi maviri kana matatu. Imwe pfungwa yakakosha ine chekuita nemavector iyuniti vector. Chinyorwa chino chichakurukura tsananguro yeyuniti vector, maitiro ekuiverenga, uye kupa mienzaniso yakati wandei yezvinetso nemhinduro.
Kunzwisisa Zvishandiso zveMayuniti
Vector yeyuniti ivector ine hukuru hweyuniti imwe chete uye divi rakafanana nevector yekutanga. Mavector eyuniti anowanzo shandiswa kurerutsa ongororo nekuti hukuru hwawo hunogara huri humwe chete, zvichiita kuti chinangwa chikuru chive padivi rawo. Kuti tishandure vector kuita vector yeyuniti, tinofanira kupatsanura chimwe nechimwe chezvikamu zvayo nehukuru hwevector.
Pamasvomhu, kana \( \mathbf{v} \) iri vector, saka unit vector yayo \( \mathbf{\hat{v}} \) inogona kuratidzwa seizvi:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}
\]
apo \( \|\mathbf{v}\| \) ndiko kukura kana kureba kwevector \( \mathbf{v} \).
Kuverenga Ukuru hweVector
Hukuru hwevector \( \mathbf{v} \) munzvimbo ine mativi maviri ine zvikamu \( (v_x, v_y) \) hunogona kuverengerwa uchishandisa fomura:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}
\]
Zvichakadaro, kune mavector ari munzvimbo ine mativi matatu ane zvikamu \( (v_x, v_y, v_z) \), hukuru hunoverengerwa uchishandisa fomura:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
Mibvunzo yemuenzaniso nekukurukurirana
Kuti tijekese pfungwa yemayuniti vectors, ngatitarisei mimwe mienzaniso yemibvunzo nehurukuro dzayo.
Muenzaniso Mubvunzo 1
Mubvunzo: Wakapihwa vhekitari \( \mathbf{a} = (3, 4) \). Sarudza vhekitari yeyuniti yevhekitari \( \mathbf{a} \).
Kukurukurirana:
1. Sarudza zvikamu zvevector \( \mathbf{a} \):
\( a_x = 3 \), \( a_y = 4 \)
2. Verenga hukuru hwevector \( \mathbf{a} \):
\[
\|\mathbf{a}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
3. Verenga vhekitari yeyuniti nekukamura chikamu chimwe nechimwe chevhekitari \( \mathbf{a} \) nehukuru hwayo:
\[
\mathbf{\hat{a}} = \left( \frac{3}{5}, \frac{4}{5} \right) = \left( 0.6, 0.8 \right)
\]
Saka, vhekitari yeyuniti ye \( \mathbf{a} \) ndi \( (0.6, 0.8) \).
Muenzaniso Mubvunzo 2
Mubvunzo: Wakapihwa vhekitari \( \mathbf{b} = (1, -2, 2) \). Sarudza vhekitari yeyuniti yevhekitari \( \mathbf{b} \).
Kukurukurirana:
1. Sarudza zvikamu zvevector \( \mathbf{b} \):
\( b_x = 1 \), \( b_y = -2 \), \( b_z = 2 \)
2. Verenga hukuru hwevhekita \( \mathbf{b} \):
\[
\|\mathbf{b}\| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
3. Verenga vhekitari yeyuniti nekukamura chikamu chimwe nechimwe chevhekitari \( \mathbf{b} \) nehukuru hwayo:
\[
\mathbf{\hat{b}} = \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \approx \left( 0.333, -0.667, 0.667 \right)
\]
Saka, vhekitari yeyuniti ye \( \mathbf{b} \) iri \( \left( 0.333, -0.667, 0.667 \right) \).
Muenzaniso Mubvunzo 3
Mubvunzo: Tichifunga nezve vector \( \mathbf{c} = (-7, 24) \). Sarudza vector yeyuniti yevector \( \mathbf{c} \).
Kukurukurirana:
1. Sarudza zvikamu zvevector \( \mathbf{c} \):
\( c_x = -7 \), \( c_y = 24 \)
2. Verenga hukuru hwevector \( \mathbf{c} \):
\[
\|\mathbf{c}\| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25
\]
3. Verenga vhekitari yeyuniti nekukamura chikamu chimwe nechimwe chevhekitari \( \mathbf{c} \) nehukuru hwayo:
\[
\mathbf{\hat{c}} = \left( \frac{-7}{25}, \frac{24}{25} \right) = \left( -0.28, 0.96 \right)
\]
Saka, vhekitari yeyuniti ye \( \mathbf{c} \) ndi \( (-0.28, 0.96) \).
Muenzaniso Mubvunzo 4
Mubvunzo: Kana vhekitari \( \mathbf{d} = (6, 8, 0) \), sarudza vhekitari yeyuniti yevhekitari \( \mathbf{d} \).
Kukurukurirana:
1. Sarudza zvikamu zvevector \( \mathbf{d} \):
\( d_x = 6 \), \( d_y = 8 \), \( d_z = 0 \)
2. Verenga hukuru hwevhekita \( \mathbf{d} \):
\[
\|\mathbf{d}\| = \sqrt{6^2 + 8^2 + 0^2} = \sqrt{36 + 64 + 0} = \sqrt{100} = 10
\]
3. Verenga vhekitari yeyuniti nekukamura chikamu chimwe nechimwe chevhekitari \( \mathbf{d} \) nehukuru hwayo:
\[
\mathbf{\hat{d}} = \left( \frac{6}{10}, \frac{8}{10}, \frac{0}{10} \right) = \left( 0.6, 0.8, 0 \right)
\]
Saka, vhekitari yeyuniti ye \( \mathbf{d} \) ndi \( (0.6, 0.8, 0) \).
Penutup
Kuburikidza nehurukuro nemienzaniso iri pamusoro apa, tinogona kunzwisisa kuti kuverenga vector yeyuniti kunoda kuverenga hukuru hwevector wobva waparadzanisa zvikamu zvevector nehukuru ihwohwo. Mavector eyuniti anobatsira zvikuru mumashandisirwo akasiyana-siyana akadai sekugadzirisa vector mumifananidzo yekombuta, kuongorora simba mufizikisi, nedzimwe nzvimbo dzakawanda. Nekunzwisisa pfungwa iyi, tinofanira kukwanisa kugadzirisa matambudziko ane chekuita nemavector zviri nyore.
