Faʻataʻitaʻiga o Faʻafitauli e Talanoaina ai Vectors Faʻafeagai
O le vector o se mea fa'amatematika e iai lona tele ma lona itu. I le su'esu'eina o vectors, e masani ona tatou fetaia'i ma vectors e iai ona uiga fa'apitoa. O se tasi o manatu autū i vectors o le inverse vector, po'o le negative vector. O lenei tusiga o le a aofia ai fa'ata'ita'iga ma se talanoaga o inverse vectors.
Malamalama i Vectors Fa'afeagai
O le vector fa'afeagai, e masani ona ta'ua o le vector leaga, o se vector e tutusa le tele ae fa'afeagai le itu ma le vector muamua. Afai o se vector e fa'ailoa mai e le \(\vec{a}\), o lona vector fa'afeagai o le \(-\vec{a}\). I le fa'amatematika, afai \(\vec{a} = (a_1, a_2, a_3)\), ona \(-\vec{a} = (-a_1, -a_2, -a_3)\).
Fa'ata'ita'iga Fesili 1
I le tuuina atu o le vector \(\vec{a} = (3, 4, -2)\). Fuafua le inverse vector o le \(\vec{a}\).
Talanoaga:
Ina ia iloa le vector fa'afeagai o le \(\vec{a}\), e na'o le pau lava le mea e mana'omia ona tatou suia o vaega ta'itasi o le vector i le leaga:
\[
-\vec{a} = (-3, -4, 2)
\]
O lea la, o le vector fa'afeagai o le \(\vec{a} = (3, 4, -2)\) o le \(-\vec{a} = (-3, -4, 2)\).
Fa'ata'ita'iga Fesili 2
Ia tuu le vector \(\vec{b} = (7, -5, 0)\). Saili le inverse vector o le \(\vec{b}\) ma faamaonia o le \(\vec{b} + (-\vec{b}) = \vec{0}\).
Talanoaga:
Muamua, matou te faʻamatalaina le vector faʻafeagai o le \(\vec{b}\):
\[
-\vec{b} = (-7, 5, 0)
\]
Sosoo ai, matou te faʻamaonia o le faʻaopoopoga o le vector \(\vec{b}\) ma lona inverse vector e mafua ai le zero vector:
\[
\vec{b} + (-\vec{b}) = (7, -5, 0) + (-7, 5, 0)
\]
Matou te faʻaopoopoina vaega o le vector:
\[
(7 – 7, -5 + 5, 0 + 0) = (0, 0, 0)
\]
O lea la, \(\vec{b} + (-\vec{b}) = \vec{0}\), ua faʻamaonia o le taunuuga o le faʻaopoopoga o le vector \(\vec{b}\) ma lona inverse vector o le zero vector lea.
Fa'ata'ita'iga Fesili 3
I le tu'uina atu o vectors \(\vec{u} = (2, -1)\) ma le \(\vec{v} = (-2, 1)\). O le \(\vec{u}\) ea le inverse vector o le \(\vec{v}\) ?
Talanoaga:
Ina ia iloa pe o \(\vec{u}\) ma le \(\vec{v}\) o ni inverse vectors, e manaʻomia ona tatou siaki pe o \(\vec{v} = -\vec{u}\).
Fuafua \(-\vec{u}\):
\[
-\vec{u} = (-2, 1)
\]
Ua iloa ai o le \(-\vec{u} = \vec{v}\), o lona uiga o le vector \(\vec{u}\) o le vector fa'afeagai moni lava lea o le vector \(\vec{v}\).
Fa'ata'ita'iga Fesili 4
Afai e iloa o le vector \(\vec{w}\) e 5 lona telē ma e fa'afeagai lona itu ma le vector \(\vec{p} = (4, 3)\), fuafua le \(\vec{w}\) i le tulaga o vaega.
Talanoaga:
Muamua, tatou te maua le tele o le vector \(\vec{p}\):
\[
|\vec{p}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
Talu ai o le \(\vec{w}\) e tutusa le telē ma le \(\vec{p}\) ae o le itu faafeagai, ona:
\[
\vec{w} = -\vec{p} = (-4, -3)
\]
O lea la, o le vector \(\vec{w}\) i le tulaga o le component o le \(\vec{w} = (-4, -3)\).
Fa'ata'ita'iga Fesili 5
Tu'uina atu le tulaga A(2, 3) ma le tulaga B(4, 7). Fuafua le vector tulaga mai le tulaga A i le tulaga B ma le vector e fa'afeagai ma lena vector.
Talanoaga:
Veketo tulaga mai le tulaga A i le tulaga B:
\[
\vec{AB} = (B_x – A_x, B_y – A_y) = (4 – 2, 7 – 3) = (2, 4)
\]
O le vector fa'afeagai o le \(\vec{AB}\):
\[
-\vec{AB} = (-2, -4)
\]
O lea la, o le vector e fa'afeagai ma le position vector \(\vec{AB} = (2, 4)\) o le \(-\vec{AB} = (-2, -4)\).
Fa'ata'ita'iga Fesili 6
Afai ua tu'uina atu se vector \(\vec{m} = (x, y)\) ma o le inverse vector o le \(\vec{m}\) o le \( (-5, 12)\). Fuafua tau o le x ma le y.
Talanoaga:
O le vector fa'afeagai o le \(\vec{m}\) o le \( (-x, -y) \), ma e tusa ai ma le fa'afitauli, \((-x, -y) = (-5, 12)\).
I le fa'afetauiina o vaega vector, tatou te maua ai:
\[
-x = -5 \fa'aalia ai le x = 5
\]
\[
-y = 12 \fa'aalia ai le y = -12
\]
O lea la, o le tau o le \(x\) e 5 ma o le tau o le \(y\) e -12.
I'uga
O vectors fa'afeagai o vectors ia e tutusa le tele ae fa'afeagai le itu. I le malamalama i le manatu o vectors fa'afeagai, e mafai ona tatou foia fa'afitauli eseese e feso'ota'i ma vectors, e pei o le fuafuaina o le leaga o se vector, fa'amaonia le fa'aopoopoina o vectors i le zero, ma isi mea fa'apena. O le talanoaina o fa'afitauli fa'ata'ita'i o lo'o i luga e fa'amoemoe e fa'ateleina ai lo tatou iloa ma le malamalama i le galulue fa'atasi ma vectors fa'afeagai.
