Faʻataʻitaʻiga o Faʻafitauli e Talanoaina ai Vectors Tulaga
O vectors o se manatu autū i le matematika ma le fisiki, e fai ma sui o aofaʻiga i itu e lua ma le tele. I faʻaoga eseese, e masani ona faʻaaogaina vectors e faʻamatalaina ai le tulaga, saoasaoa, malosi, ma le tele o isi faʻasologa. I totonu o ituaiga eseese o vectors, e taua tele le sao o vectors tulaga i le faʻafanua o le tulaga o se nofoaga i le vateatea.
Fa'amatalaga o le Vector Tulaga
O le vector tulaga o se vector e faʻamatalaina ai le tulaga o se tulaga e faʻatatau i le amataga i totonu o se faiga faʻamaopoopo. I se tulaga lautele, o se vector tulaga e tusia i le faʻatulagaga faʻamaopoopo Cartesian e pei ona:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
O iinei, o le \(\mathbf{r}\) o le vector tulaga, o le \(x\), o le \(y\), ma le \(z\) o ona vaega i luga o le \(x\), \(y\), ma le \(z\) axes, i le faasologa lava lea, ae o le \(\mathbf{i}\), o le \(\mathbf{j}\), ma le \(\mathbf{k}\) o vectors iunite e tutusa ma le coordinate axes, i le faasologa lava lea. I le avanoa lua-dimensional, e masani lava ona leai le vaega \(z\), o lea e avea ai le vector tulaga:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} \]
Talosaga Vector Tulaga
Mo se faʻataʻitaʻiga, i le fisiki, e taua tele le sao o vectors o tulaga i le faʻamatalaina o le gaioiga o mea faitino. O le tulaga o se mea faitino e faʻatatau i le amataga (point reference) e mafai ona faʻatusalia e se vector o tulaga. E le gata i lea, i le inisinia faʻainisinia, o le faʻatatauina o malosiaga ma moments e masani ona aofia ai le faʻaaogaina o vectors o tulaga.
Fesili Fa'ata'ita'i ma Talanoaga o Vectors Tulaga
Fesili 1
Faapea o loo i ai ni togi se lua i le avanoa 3D, o le togi A ma ni faamaopoopoga \( (1, 2, 3) \) ma le togi B ma ni faamaopoopoga \( (4, 0, -2) \). Fuafua ia vectors o tulaga o togi A ma le B. E le gata i lea, fuafua le vector e feso'ota'i ai le togi A i le togi B.
Talanoaga:
Vector tulaga mo le tulaga A:
\[ \mathbf{r_A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \]
Vector tulaga mo le tulaga B:
\[ \mathbf{r_B} = 4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k} \]
Sosoo ai, ina ia maua le vector e fesoʻotaʻi ai le point A i le point B (e taʻua o le \(\mathbf{AB}\)), e manaʻomia ona tatou toʻesea le position vector o le A mai le position vector o le B:
\[ \mathbf{AB} = \mathbf{r_B} – \mathbf{r_A} \]
O lea la, sui ia vectors tulaga e lua o loʻo i luga:
\[ \mathbf{AB} = (4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k}) – (1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \]
\[ \mathbf{AB} = (4 – 1)\mathbf{i} + (0 – 2)\mathbf{j} + (-2 – 3)\mathbf{k} \]
\[ \mathbf{AB} = 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \]
O lea la, o le vector e fesoʻotaʻi ai le point A i le B o le \( 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \).
Fesili 2
Afai o lo'o i luga o le \((2, 3)\) se tulaga P i le va'alele 2D, saili le umi (masani) o le vector tulaga \(\mathbf{r_P}\).
Talanoaga:
Vector tulaga o le tulaga P:
\[ \mathbf{r_P} = 2\mathbf{i} + 3\mathbf{j} \]
E mafai ona fuafuaina le umi o le vector tulaga \(\mathbf{r_P}\) e faʻaaoga ai le fua faʻatatau o le vector norm (poʻo le umi):
\[ \| \mathbf{r_P} \| = \sqrt{x^2 + y^2} \]
Suitulaga ia tau o le \(x\) ma le \(y\):
\[ \| \mathbf{r_P} \| = \sqrt{2^2 + 3^2} \]
\[ \| \mathbf{r_P} \| = \sqrt{4 + 9} \]
\[ \| \mathbf{r_P} \| = \sqrt{13} \]
O lea la, o le umi o le vector tulaga \(\mathbf{r_P}\) o le \(\sqrt{13}\).
Fesili 3
Faapea o loo taoto se tulaga Q i le \( (5, -4, 2) \). Saili le tulimanu i le va o le vector tulaga \(\mathbf{r_Q}\) ma le \(x\) axis.
Talanoaga:
Vector tulaga o le tulaga Q:
\[ \mathbf{r_Q} = 5\mathbf{i} – 4\mathbf{j} + 2\mathbf{k} \]
Mo le sailia o le tulimanu i le va o le vector \(\mathbf{r_Q}\) ma le \(x\) axis, e mafai ona tatou faʻaaogaina le manatu o le dot product. Muamua, tatou te fuafuaina le dot product i le va o le \(\mathbf{r_Q}\) ma le \(\mathbf{i}\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5\mathbf{i} \cdot \mathbf{i} + (-4\mathbf{j} \cdot \mathbf{i}) + 2\mathbf{k} \cdot \mathbf{i} \]
Talu ai o le \(\mathbf{i} \cdot \mathbf{i} = 1\), \(\mathbf{j} \cdot \mathbf{i} = 0\), ma le \(\mathbf{k} \cdot \mathbf{i} = 0\), ona:
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5 \]
Tulaga masani o le \(\mathbf{r_Q}\):
\[ \| \mathbf{r_Q} \| = \sqrt{5^2 + (-4)^2 + 2^2} \]
\[ \| \mathbf{r_Q} \| = \sqrt{25 + 16 + 4} \]
\[ \| \mathbf{r_Q} \| = \sqrt{45} \]
\[ \| \mathbf{r_Q} \| = 3\sqrt{5} \]
O le tulaga masani o le \(\mathbf{i}\) e 1, auā o le \(\mathbf{i}\) o se vector iunite.
Fa'aaogaina o le fua fa'atatau o le dot product e su'e ai le tulimanu \(\theta\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = \| \mathbf{r_Q} \| \| \mathbf{i} \| \cos\theta\]
\[ 5 = 3\sqrt{5} \cos\theta \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \]
\[ \cos\theta = \frac{5\sqrt{5}}{15} \]
\[ \cos\theta = \frac{\sqrt{5}}{3} \]
O le tulimanu \(\theta\) i le va o le position vector \(\mathbf{r_Q}\) ma le \(x\) axis e faapea:
\[ \theta = \cos^{-1} \left(\frac{\sqrt{5}}{3}\right) \]
I'uga
E taua tele le sao o vectors o tulaga i le saienisi ma le inisinia, aemaise lava i le fa'afanua o le tulaga o mea faitino i le avanoa fa'amaopoopo. O fa'ata'ita'iga o lo'o i luga o lo'o fa'aalia ai le auala e fuafua ai vectors o tulaga, o latou umi, ma tulimanu i le va o latou ma le coordinate axes. O le malamalama i nei manatu fa'avae e matua'i taua tele i le fo'ia o fa'afitauli eseese e aofia ai le avanoa ma fa'amaopoopo i le matematika ma le fisiki.
