O vectors o se manatu taua i le fisiki, e faʻaaogaina e fai ma sui o aofaʻiga i le tele ma le itu. I le fisiki, e masani ona faʻaaogaina vectors e faʻamatalaina ai mea eseese e pei o le malosi, saoasaoa, faʻavavevaveina, ma isi mea. O lenei tusiga o le a talanoaina ai ni faʻataʻitaʻiga o faʻafitauli o vectors fisiki, faʻatasi ai ma a latou fofo ma faʻamatalaga.
1. Fa'aopoopoga ma To'esega o le Vector
Fa'ata'ita'iga Fesili 1:
E lua vectors \(\mathbf{A}\) ma le \(\mathbf{B}\) ua tu'uina atu e pei ona taua i lalo:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Fuafua:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Tali:
Mo le faaopoopoina o vectors e lua, matou te faaopoopoina o la'ua vaega eseese.
1. \(\mathbf{A} + \mathbf{B}\):
\[
\mathbf{A} + \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) + (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – 2)\mathbf{i} + (4 + 5)\mathbf{j}
\]
\[
= 1\mathbf{i} + 9\mathbf{j}
\]
\[
\mathbf{A} + \mathbf{B} = \mathbf{i} + 9\mathbf{j}
\]
2. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
O lea la, o le taunuuga:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Fa'atelega Fa'asolosolo (Dot Product)
Fa'ata'ita'iga Fesili 2:
E lua vectors \(\mathbf{C}\) ma le \(\mathbf{D}\) ua tu'uina atu e pei ona taua i lalo:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Fuafua le fua fa'atatau (fua fa'atatau o le togitogi) o le \(\mathbf{C}\) ma le \(\mathbf{D}\).
Tali:
O le fua fa'atatau o vectors e lua \(\mathbf{C}\) ma le \(\mathbf{D}\) e fa'apea:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
O lea la, o le iʻuga o le oloa faʻasolosolo (scalar product) o le \(\mathbf{C}\) ma le \(\mathbf{D}\) e 26.
3. Oloa Fa'alava
Fa'ata'ita'iga Fesili 3:
E lua vectors \(\mathbf{E}\) ma le \(\mathbf{F}\) ua tu'uina atu e pei ona taua i lalo:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Fuafua le fa'asologa o le \(\mathbf{E}\) ma le \(\mathbf{F}\).
Tali:
E mafai ona fuafuaina le fa'asologa o le oloa (cross product) o vectors e lua \(\mathbf{E}\) ma le \(\mathbf{F}\) e fa'aaoga ai le matrix determinant:
\[
\mathbf{E} \taimi \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4&5&6
\end{vmatrix}
\]
Fuafua le mea e fuafua ai le matrix:
\[
\mathbf{E} \times \mathbf{F} = \mathbf{i} (2 \cdot 6 – 3 \cdot 5) – \mathbf{j} (1 \cdot 6 – 3 \cdot 4) + \mathbf{k} (1 \cdot 5 – 2 \cdot 4)
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
O lea la, o le taunuuga o le oloa fa'alava o le \(\mathbf{E}\) ma le \(\mathbf{F}\) e faapea:
\[
\mathbf{E} \taimi \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Tele o le Vekita
Fa'ata'ita'iga Fesili 4:
I le tuuina atu o le vector \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Fuafua le tele (umi) o le vector \(\mathbf{G}\).
Tali:
E mafai ona fuafuaina le telē o le vector \(\mathbf{G}\) e faʻaaoga ai le fua faʻatatau:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
O lea la, o le telē o le vector \(\mathbf{G}\) e 5.
5. Fa'ai'uga o le Veka
Fa'ata'ita'iga Fesili 5:
O le vector \(\mathbf{H}\) e 10 iunite le telē ma e fausia ai se tulimanu e 30° ma le x-axis. Fuafua vaega o le vector \(\mathbf{H}\) i luga o le x- ma le y-axis.
Tali:
E mafai ona fuafuaina vaega o le vector \(\mathbf{H}\) i luga o le x (\(\mathbf{H}_x\)) ma le y (\(\mathbf{H}_y\)) axes e fa'aaoga ai le trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \sala(\theta)
\]
Fa'atasi ai ma le \(|\mathbf{H}| = 10\) ma le \(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \sin(30°)
\]
O tau o le \(\cos(30°) = \frac{\sqrt{3}}{2}\) ma le \(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
O lea la, o vaega o le vector \(\mathbf{H}\) e faapea:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
I'uga
I totonu o lenei tusiga, ua matou talanoaina ai ni faʻataʻitaʻiga o faʻafitauli e aofia ai vectors i le fisiki, e amata mai i le faʻaopoopoina ma le toʻesea o vectors, faʻatelega scalar ma le cross, i le tele ma le iugafono o vectors. O le malamalama i le manatu ma le faʻagaioiga o vectors e taua tele i le fisiki aua e tele mea faʻalenatura e mafai ona faʻamatalaina e faʻaaoga ai vectors. E faʻamoemoe o nei faʻataʻitaʻiga o faʻafitauli o le a fesoasoani ia te oe e malamalama atili ai i le manatu o vectors.
