Fa'ata'ita'iga o Fesili ma Talanoaga o le To'esega o Vector
Pendahuluan
I le matematika ma le fisiki, o vectors o se manatu faavae e faʻaaogaina e faʻamatalaina ai le tele o mea faʻalenatura ma mea faʻainisinia. O se vector o se aofaʻiga e iai le tele ma le itu e agaʻi i ai. O nisi o faʻataʻitaʻiga taua o vectors o le displacement, velocity, acceleration, ma le force. I totonu o lenei tusiga, o le a tatou talanoaina le vector subtraction, e ui o lenei autu e masani ona faʻamamafaina i le anotusi o le vector combination.
O le to'esea o vector o se fa'agaioiga taua tele i le su'esu'ega vector. Mo se su'esu'ega loloto i lenei manatu, se'i o tatou toe iloiloina ni fa'ata'ita'iga o fa'afitauli ma talanoaga e feso'ota'i ma le to'esea o vector.
To'esega Vekita
O le vector to'esega {\displaystyle \mathbf{A} – \mathbf{B}} ua fa'amatalaina o le fa'agaioiga o le fa'aopoopoina o le vector {\displaystyle \mathbf{A}} fa'atasi ai ma le vector {\displaystyle -\mathbf{B}}, lea o le {\displaystyle -\mathbf{B}} o se vector e tutusa le tele ma le {\displaystyle \mathbf{B}} ae fa'afeagai le itu. I le fa'amatematika, e mafai ona tusia lenei mea e pei o:
{\displaystyle \mathbf{A} – \mathbf{B} = \mathbf{A} + (-\mathbf{B})}
Fesili Fa'ata'ita'i ma Talanoaga
Fesili 1: To'esea o Vectors Lua-Fua
Faapea o loo i ai ni vectors se lua i totonu o faamaopoopoga Cartesian:
{\displaystyle \mathbf{A} = (4, 3)} ma le {\displaystyle \mathbf{B} = (1, 2)}. Fa'atatau {\displaystyle \mathbf{A} – \mathbf{B}}.
Talanoaga:
O le laasaga muamua o le sailia lea o le vector negative o le {\displaystyle \mathbf{B}}, e pei o:
{\displaystyle -\mathbf{B} = (-1, -2)}
Sosoo ai, faaopoopo le vector {\displaystyle \mathbf{A}} faatasi ai ma le {\displaystyle -\mathbf{B}}:
{\displaystyle \mathbf{A} – \mathbf{B} = (4, 3) + (-1, -2)}
Fa'atino le fa'aopoopoga vector e ala i le fa'aopoopoina o vaega ta'itasi o le x ma le y:
{\displaystyle \mathbf{A} – \mathbf{B} = (4 + (-1), 3 + (-2))}
{\displaystyle \mathbf{A} – \mathbf{B} = (3, 1)}
O lea la, o le taunuuga o le to'esea o vectors {\displaystyle \mathbf{A} – \mathbf{B}} o le vector (3, 1).
Fesili 2: To'esea o Vectors e Tolu-Fua
I le tuuina atu o ni vectors se lua i ni coordinates e tolu-dimensional:
{\displaystyle \mathbf{P} = (2, -4, 6)} ma le {\displaystyle \mathbf{Q} = (-3, 5, 7)}. Fa'atatau {\displaystyle \mathbf{P} – \mathbf{Q}}.
Talanoaga:
O le laasaga muamua o le sailia lea o le vector leaga o le {\displaystyle \mathbf{Q}}:
{\displaystyle -\mathbf{Q} = (3, -5, -7)}
Sosoo ai, faaopoopo le vector {\displaystyle \mathbf{P}} faatasi ai ma le {\displaystyle -\mathbf{Q}}:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2, -4, 6) + (3, -5, -7)}
Fa'atino le fa'aopoopoga vector e ala i le fa'aopoopoina o vaega ta'itasi o le x, y, ma le z:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2 + 3, -4 + (-5), 6 + (-7))}
{\displaystyle \mathbf{P} – \mathbf{Q} = (5, -9, -1)}
O lea la, o le taunuuga o le to'esea o vectors {\displaystyle \mathbf{P} – \mathbf{Q}} o le vector (5, -9, -1).
Fesili 3: To'esega Vector i le Va'alele Faigata
Faapea o loo i ai ni vectors se lua o loo faatusalia e numera faigata:
{\displaystyle \mathbf{M} = 3 + 4i} ma le {\displaystyle \mathbf{N} = 1 + 2i}. Fa'atatau {\displaystyle \mathbf{M} – \mathbf{N}}.
Talanoaga:
O le laasaga muamua o le sailia lea o le vector leaga o le {\displaystyle \mathbf{N}}:
{\displaystyle -\mathbf{N} = -1 – 2i}
Sosoo ai, faaopoopo le vector {\displaystyle \mathbf{M}} faatasi ai ma le {\displaystyle -\mathbf{N}}:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + 4i) + (-1 – 2i)}
Fa'atino le fa'aopoopoga vector e ala i le fa'aopoopoina o vaega moni ma vaega fa'ata'ita'i ta'itasi:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + (-1)) + (4i + (-2i))}
{\displaystyle \mathbf{M} – \mathbf{N} = 2 + 2i}
O lea la, o le taunuuga o le to'esea o vectors {\displaystyle \mathbf{M} – \mathbf{N}} o le numera lavelave 2 + 2i.
Fesili 4: To'esega o Vector i le Polar Coordinate System
Faapea o loo i ai ni vectors se lua i totonu o coordinates polar:
{\displaystyle \mathbf{U}} e 5 lona telē ma e 30° lona tulimanu,
ma o le {\displaystyle \mathbf{V}} e 3 lona telē ma e 150° lona tulimanu.
Fuafua {\displaystyle \mathbf{U} – \mathbf{V}}.
Talanoaga:
O le laasaga muamua o le liua lea o vectors {\displaystyle \mathbf{U}} ma le {\displaystyle \mathbf{V}} i fa'amaopoopoga Cartesian.
Mo {\displaystyle \mathbf{U}}:
{\displaystyle U_x = 5 \cos(30^\circ) = 5 \left(\frac{\sqrt{3}}{2}\right) = 5 \cdot 0.866 = 4.33}
{\displaystyle U_y = 5 \sin(30^\circ) = 5 \left(\frac{1}{2}\right) = 5 \cdot 0.5 = 2.5}
O lea la o le {\displaystyle \mathbf{U}} i le Cartesian o le (4.33, 2.5).
Mo {\displaystyle \mathbf{V}}:
{\displaystyle V_x = 3 \cos(150^\circ) = 3 \left(\frac{-\sqrt{3}}{2}\right) = 3 \cdot (-0.866) = -2.598}
{\displaystyle V_y = 3 \sin(150^\circ) = 3 \left(\frac{1}{2}\right) = 3 \cdot 0.5 = 1.5}
O lea la o le {\displaystyle \mathbf{V}} i le Cartesian o le (-2.598, 1.5).
Laasaga e sosoo ai, fuafua le vector subtraction i le Cartesian:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33, 2.5) – (-2.598, 1.5)}
O lona uiga o le faaopoopoina o le leaga o le vector:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33 + 2.598, 2.5 – 1.5)}
{\displaystyle \mathbf{U} – \mathbf{V} = (6.928, 1)}
O lea la, o le taunuuga o le to'esea o le vector {\displaystyle \mathbf{U} – \mathbf{V}} i fa'amaopoopoga Cartesian o le (6.928, 1).
I'uga
O le to'esea o vector o se fa'agaioiga fa'amatematika taua i le tele o matā'upu e fa'aaogaina ai le au'ili'iliga vector. Pe o faiga fa'amaopoopo lua-dimensional, tolu-dimensional, lavelave, po'o polar coordinate, e tumau pea le mataupu faavae autu: fa'aopoopoina o le tasi vector i le negative o le isi. O fa'ata'ita'iga o lo'o i luga o lo'o fa'aalia ai auala eseese e fa'aoga ai lenei fa'agaioiga i tulaga eseese, e fesoasoani ia i tatou e malamalama atili ai i le manatu ma le fa'atinoina.
