Fa'ata'ita'iga o Fesili e Talanoaina ai Vectors Leaga po'o Vectors Fa'afeagai
I le matematika, aemaise lava i le fisiki poʻo le geometry analytical, o le manatu o vectors e taua tele. E masani ona faʻaaogaina vectors e fai ma sui o aofaʻiga i itu uma e lua ma le tele, e pei o le saoasaoa, malosi, ma le fesiitaiga. Pe a talanoaina vectors, e masani ona tatou fetaiaʻi ma faaupuga "negative vector" poʻo le "opposite vector." O lenei tusiga o le a faʻamatalaina loloto ai lenei manatu, ma tuʻuina atu faʻataʻitaʻiga ma fofo e faʻafaigofie ai le malamalama.
Fa'amatalaga o le Vector Negative
O se vector leaga, poʻo se vector faʻafeagai, o se vector e faʻafeagai lona itu ae tutusa le tele ma le vector muamua. Afai e iai sa tatou vector \(\mathbf{a}\), o lona uiga o le vector leaga o le \(\mathbf{a}\), e masani ona faʻailoaina o le \(-\mathbf{a}\), e faʻafeagai lona itu ma tutusa le tele ma le \(\mathbf{a}\). Afai o le \(\mathbf{a}\) o loʻo faʻatusalia i le foliga o le \((a_x, a_y)\), o lona uiga o le vector leaga o le \((-a_x, -a_y)\).
Fa'ailoga ma le Fa'atusa o Vector
Faapea o se vector \(\mathbf{a}\) o loʻo faʻatusalia i le tulaga o vaega e pei o:
\[ \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} \]
lea o le \(\mathbf{i}\) ma le \(\mathbf{j}\) o vectors iunite i le x- ma le y-directions, i le faasologa. Ona, o le vector leaga \(\mathbf{a}\) po o le \(-\mathbf{a}\) e mafai ona fa'atusalia e pei o:
\[ -\mathbf{a} = -a_x \mathbf{i} – a_y \mathbf{j} \]
Uiga o Vectors Negative
O nisi o meatotino taua o vectors leaga e aofia ai:
1. Fa'aopoopoga i le Vector Muamua: O le fa'aopoopoina o se vector ma lona vector leaga o le a maua ai se vector zero.
\[ \mathbf{a} + (-\mathbf{a}) = \mathbf{0} \]
2. Fa'agaioiga Fa'asolosolo: O le fa'ateleina o se vector i le -1 o le a maua ai lona vector leaga.
\[ -1 \cdot \mathbf{a} = -\mathbf{a} \]
Fesili Fa'ata'ita'i ma Talanoaga
Ina ia malamalama atili i le manatu o vectors leaga poʻo vectors feteenai, seʻi o tatou galulue i faʻataʻitaʻiga faʻafitauli nei:
Faataitaiga 1:
Faapea o loo i ai se vector \(\mathbf{a} = 3 \mathbf{i} – 4 \mathbf{j}\). Fuafua le vector leaga o le vector \(\mathbf{a}\).
Talanoaga:
Ua iloa:
\[ \mathbf{a} = 3 \mathbf{i} – 4 \mathbf{j} \]
O le vector leaga o le \(\mathbf{a}\) e faapea:
\[ -\mathbf{a} = -1 \cdot (3 \mathbf{i} – 4 \mathbf{j}) \]
\[ -\mathbf{a} = -3 \mathbf{i} + 4 \mathbf{j} \]
O lea la, o le vector leaga o le \(\mathbf{a}\) e faapea:
\[ -\mathbf{a} = -3 \mathbf{i} + 4 \mathbf{j} \]
Faataitaiga 2:
E lua veki \(\mathbf{b} = 6 \mathbf{i} + 2 \mathbf{j}\) ma \(\mathbf{c} = -1 \mathbf{i} + 7 \mathbf{j}\). Su'e le fua o le \(\mathbf{b} + (-\mathbf{c})\).
Talanoaga:
Ua iloa:
\[ \mathbf{b} = 6 \mathbf{i} + 2 \mathbf{j} \]
\[ \mathbf{c} = -1 \mathbf{i} + 7 \mathbf{j} \]
O le vector leaga o le \(\mathbf{c}\) e faapea:
\[ -\mathbf{c} = -1 \cdot (-1 \mathbf{i} + 7 \mathbf{j}) \]
\[ -\mathbf{c} = 1 \mathbf{i} – 7 \mathbf{j} \]
O lea ua tatou maua \(\mathbf{b} + (-\mathbf{c})\):
\[ \mathbf{b} + (-\mathbf{c}) = (6 \mathbf{i} + 2 \mathbf{j}) + (1 \mathbf{i} – 7 \mathbf{j}) \]
\[ \mathbf{b} + (-\mathbf{c}) = (6 + 1) \mathbf{i} + (2 – 7) \mathbf{j} \]
\[ \mathbf{b} + (-\mathbf{c}) = 7 \mathbf{i} – 5 \mathbf{j} \]
O lea la, o le taunuuga o le \(\mathbf{b} + (-\mathbf{c})\) e faapea:
\[ 7 \mathbf{i} – 5 \mathbf{j} \]
Faataitaiga 3:
O lo'o iai se ve'a \(\mathbf{d} = a \mathbf{i} + b \mathbf{j}\), o le a ma le b o numera moni. Afai \(\mathbf{d} + \mathbf{e} = \mathbf{0}\), fuafua le vete \(\mathbf{e}\).
Talanoaga:
Ua iloa:
\[ \mathbf{d} = a \mathbf{i} + b \mathbf{j} \]
\[ \mathbf{d} + \mathbf{e} = \mathbf{0} \]
Ina ia maua le \(\mathbf{e}\), e mafai ona tatou tusia:
\[ \mathbf{e} = -\mathbf{d} \]
O lea la, o le vector \(\mathbf{e}\) o le vector leaga lea o le \(\mathbf{d}\):
\[ \mathbf{e} = -\mathbf{d} = -a \mathbf{i} – b \mathbf{j} \]
Faataitaiga 4:
Tuuina atu le vete \(\mathbf{f} = 5 \mathbf{i} + k \mathbf{j}\). Ua iloa o le vete leaga o \(\mathbf{f}\) o le \(-5 \mathbf{i} – 8 \mathbf{j}\). Fuafua le tau o le k.
Talanoaga:
Ua iloa:
\[ \mathbf{f} = 5 \mathbf{i} + k \mathbf{j} \]
\[ -\mathbf{f} = -5 \mathbf{i} – 8 \mathbf{j} \]
Mai lenei mafutaga, e mafai ona tatou fausia ni fa'atusatusaga vaega mo \(\mathbf{f}\) ma le \(-\mathbf{f}\). I vaega fa'apitoa, o le vector \(\mathbf{f}\) ma lona vector leaga e tatau ona i ai le mafutaga tulaga tutusa ma fa'ailoga fa'afeagai. O lea la:
Mo vaega \( \mathbf{i} \):
\[ -5 = -5 \]
E otometi lava ona moni lenei mea.
Mo le vaega \( \mathbf{j} \):
\[ -k = -8 \]
\[ k = 8 \]
O lea la, o le tau o le \(k \) e 8.
I'uga
O le malamalama i le manatu o se vector leaga, poʻo se vector faʻafeagai, e taua tele i le suʻesuʻeina o vectors. O se vector faʻafeagai o se vector e faʻafeagai le itu ma le vector muamua ae e tutusa le tele. I galuega faʻatino vector, o le iloaina ma le faʻaaogaina o vectors leaga e mafai ona fesoasoani tele i le faʻafaigofieina o le tele o faʻafitauli, e pei o le faʻaopoopoina poʻo le toʻesea o vectors. Faatasi ai ma le faʻataʻitaʻi ma le malamalama i meatotino autu o vectors, o le malamalama i lenei manatu o le a sili atu ona faigofie ona malamalama i ai.
Matou te faʻamoemoe o fesili faʻataʻitaʻi ma talanoaga o loʻo tuʻuina atu i lenei tusiga o le a fesoasoani ia te oe e maua ai se malamalamaga loloto i vectors leaga, poʻo vectors feteenai. Ia faʻaauau pea ona faʻataʻitaʻi ma suʻesuʻe nisi fesili ina ia sili atu lou tomai i lenei mataupu!
