Mibvunzo Yemuenzaniso Yekukurukura Mavector Akaenzana muCartesian Coordinate System
Pendauluan
Mumasvomhu, vector chinhu chine hukuru uye gwara. Vectors vane mashandisirwo muzvikamu zvakasiyana-siyana zvakaita sefizikisi, uinjiniya, uye sainzi yemakombiyuta. Muchinyorwa chino, tichakurukura pfungwa yemavectors akaenzana muCartesian coordinate system uye tichapa mienzaniso nemhinduro. Kunzwisisa mavectors akaenzana kwakakosha mukushandiswa kwakasiyana-siyana, kusanganisira mechanics uye computer graphics.
Nheyo dzeVectors muCartesian Coordinate System
Sisitimu yeCartesian coordinate system ine mativi maviri ine ma axes eX neY akatarisana. Musystem iyi, ma vector anowanzo miririrwa sema pairs akarongwa (x, y), apo x ne y zviri zvikamu zve vector pamwe chete ne X ne Y axes, zvichiteerana.
Ngatitii tine mapoinzi maviri muCartesian coordinate system, \(A(x_1, y_1)\) uye \(B(x_2, y_2)\). Vector inobatanidza mapoinzi maviri aya inogona kuratidzwa se \( \vec{AB} = (x_2 – x_1, y_2 – y_1) \).
Maveji Akaenzana
Mavector maviri anonzi akaenzana kana aine hukuru uye gwara rakafanana. Pamasvomhu, mavector maviri \( \vec{u} = (u_1, u_2) \) uye \( \vec{v} = (v_1, v_2) \) akaenzana kana uye chete kana:
\[
\vec{u} = \vec{v} \quad \text{or} \quad (u_1 = v_1 \text{ na } u_2 = v_2)
\]
Izvi zvinoreva kuti zvikamu zvinoenderana zvemavector maviri zvinofanira kunge zvakafanana.
Mibvunzo yemuenzaniso nekukurukurirana
Mubvunzo 1: Kuona Mavekitori Akaenzana
Zvichipiwa mapoinzi matatu muCartesian coordinate system: \( A(2, 3) \), \( B(5, 7) \), uye \( C(7, -1) \). Sarudza kana vector \( \vec{AB} \) yakaenzana nevector \( \vec{AC} \).
Kukurukurirana:
– Sarudza vhekitari \( \vec{AB} \):
\[
\vec{AB} = (5 – 2, 7 – 3) = (3, 4)
\]
– Sarudza vhekitari \( \vec{AC} \):
\[
\vec{AC} = (7 – 2, -1 – 3) = (5, -4)
\]
Mushure mekuverenga zvikamu zvevector yega yega, tinoona kuti \( \vec{AB} = (3, 4) \) uye \( \vec{AC} = (5, -4) \). Sezvo \( (3, 4) \neq (5, -4) \), vector \( \vec{AB} \) haina kuenzana nevector \( \vec{AC} \).
Mubvunzo 2: Kugadzira Maveki Akaenzana
Sarudza poindi \( D \) zvekuti vhekitari \( \vec{AB} = \vec{CD} \) ine poindi \( C(4, -2) \), poindi \( B(8, 3) \), uye \( A(2, 1) \).
Kukurukurirana:
– Sarudza vhekitari \( \vec{AB} \):
\[
\vec{AB} = (8 – 2, 3 – 1) = (6, 2)
\]
Sezvo \( \vec{CD} \) ichifanira kunge yakaenzana ne \( \vec{AB} \), saka:
\[
\vec{CD} = \vec{AB} = (6, 2)
\]
– Ngatitii \( D(x, y) \). Zvadaro \( \vec{CD} = (x – 4, y + 2) \). Kubva pano tinowana:
\[
(x – 4, y + 2) = (6, 2)
\]
Nekuenzanisa zvikamu zvakakodzera, tinowana:
\[
x – 4 = 6 \quad \Rightarrow \quad x = 10
\]
\[
y + 2 = 2 \quad \Rightarrow \quad y = 0
\]
Saka, pfungwa \( D \) ndi \( (10, 0) \).
Mubvunzo 3: Humbowo neVector Magnitude
Ratidza kuti mavector \( \vec{PQ} \) uye \( \vec{RS} \) akaenzana, zvichipiwa \( P(1, 2) \), \( Q(4, 6) \), \( R(-3, -7) \), uye \( S(0, -3) \).
Kukurukurirana:
– Sarudza vhekitari \( \vec{PQ} \):
\[
\vec{PQ} = (4 – 1, 6 – 2) = (3, 4)
\]
– Tsanangura vhekitari \( \vec{RS} \):
\[
\vec{RS} = (0 – (-3), -3 – (-7)) = (3, 4)
\]
Kubva pamhedzisiro yekuverenga, tinoona kuti \( \vec{PQ} = (3, 4) \) uye \( \vec{RS} = (3, 4) \). Sezvo mavector ese ari maviri aine zvikamu zvakafanana, \( \vec{PQ} \) yakaenzana ne \( \vec{RS} \).
Kushandiswa kweEquivalent Vectors
Mavector akaenzana anowanzo shandiswa muzvidzidzo zvakasiyana-siyana zvesainzi. Mufizikisi, anoshandiswa kutsanangura masimba kana kutama kune hukuru negwara rakafanana. Mumifananidzo yemakombiyuta, mavector anoshandiswa kushandura uye kuratidza zvinhu zvemifananidzo zvinobudirira.
Mhedziso
Kunzwisisa pfungwa yemavector akaenzana muCartesian coordinate system ndiyo hwaro hwakakosha hwemasvomhu nemashandisirwo ayo akafara. Chinyorwa chino chakurukura maitirwo ekuona mavector akaenzana kuburikidza nematambudziko akati wandei emuenzaniso uye mhinduro dzawo. Nekunzwisisa nekushandisa pfungwa iyi, tinogona kugadzirisa matambudziko akasiyana-siyana anosanganisira kuongorora vector muminda yakawanda yesainzi.
Tinovimba kuti hurukuro iyi ichakubatsira kunzwisisa pfungwa yemavector akaenzana muCartesian coordinate system. Kudzidza kwakanaka, uye rombo rakanaka mukudzidzira mavector!
