Fa'ata'ita'iga o Fesili ma Talanoaga o Vectors e Tolu-Fua i le Cartesian Coordinate System
O vectors e tolu-vaega o se manatu taua i le matematika ma le fisiki, e masani ona faʻaaogaina e fai ma sui o mea faitino poʻo mea tutupu i le avanoa e tolu-vaega. I le faiga faʻamaopoopo Cartesian, o nei vectors e faʻatusalia e vaega e tolu, e masani ona faʻailoaina o \( (x, y, z) \). O lenei tusiga o le a talanoaina ai ni faʻataʻitaʻiga o faʻafitauli ma fofo e fesoʻotaʻi ma vectors e tolu-vaega i le faiga faʻamaopoopo Cartesian.
Malamalama i Vectors e Tolu-Fua
E mafai ona fa'aalia se vector i le avanoa tolu-vaega e pei o le \(\mathbf{A} = (A_x, A_y, A_z)\), lea:
– O le \(A_x\) o le vaega o le vector i luga o le x-axis.
– O le \(A_y\) o le vaega vector i luga o le y-axis.
– O le \(A_z\) o le vaega vector i luga o le z-axis.
Fesili Fa'ata'ita'i ma Talanoaga
Fesili 1: Fa'agaioiga Fa'aopoopo Vector
Afai e tu'uina atu ni vectors se lua, \(\mathbf{A} = (2, -3, 4)\) ma le \(\mathbf{B} = (-1, 5, 2)\). Fa'atatau le aofa'i o nei vectors e lua.
Talanoaga:
O le fa'aopoopoga o vectors e lua \(\mathbf{A}\) ma le \(\mathbf{B}\) e faia e ala i le fa'aopoopoina o a la vaega talafeagai. O lea la, ua ia i tatou:
\[
\mathbf{C} = \mathbf{A} + \mathbf{B} = (A_x + B_x, A_y + B_y, A_z + B_z)
\]
Suitulaga i tau vector ua tuʻuina atu:
\[
\mathbf{C} = (2 + (-1), -3 + 5, 4 + 2) = (1, 2, 6)
\]
O lea la, o le taunuuga o le faaopoopoina o vectors \(\mathbf{A}\) ma le \(\mathbf{B}\) o le \(\mathbf{C} = (1, 2, 6)\).
Fesili 2: Fa'agaioiga o le Vector To'esega
Tuuina atu ni vectors se lua, \(\mathbf{A} = (4, 1, -2)\) ma le \(\mathbf{B} = (5, -3, 6)\). Fuafua le to'esega o nei vectors e lua, o lona uiga \(\mathbf{A} – \mathbf{B}\).
Talanoaga:
O le toesea o vectors e lua \(\mathbf{A}\) ma le \(\mathbf{B}\) e faia e ala i le toesea o a la vaega e fetaui. O lea la, ua ia i tatou:
\[
\mathbf{D} = \mathbf{A} – \mathbf{B} = (A_x – B_x, A_y – B_y, A_z – B_z)
\]
Suitulaga i tau vector ua tuʻuina atu:
\[
\mathbf{D} = (4 – 5, 1 – (-3), -2 – 6) = (-1, 4, -8)
\]
O lea la, o le taunuuga o le to'esea o vectors \(\mathbf{A}\) ma le \(\mathbf{B}\) o le \(\mathbf{D} = (-1, 4, -8)\).
Fesili 3: Fa'agaioiga Fa'atelega Skalar
Afai ua tu'uina atu se vector \(\mathbf{A} = (3, -2, 7)\) ma se scalar \(k = 4\). Fa'atatau le scalar product o nei vectors.
Talanoaga:
O le fa'ateleina o se scalar \(k\) i se vector \(\mathbf{A}\) e faia e ala i le fa'ateleina o vaega ta'itasi o le vector i lena scalar. O lea la, ua ia i tatou:
\[
\mathbf{E} = k \cdot \mathbf{A} = k \cdot (A_x, A_y, A_z) = (k \cdot A_x, k \cdot A_y, k \cdot A_z)
\]
Suitulaga i tau o loʻo tuʻuina atu:
\[
\mathbf{E} = 4 \cdot (3, -2, 7) = (4 \cdot 3, 4 \cdot -2, 4 \cdot 7) = (12, -8, 28)
\]
O lea la, o le taunuuga o le faateleina o le scalar \(k\) i le vector \(\mathbf{A}\) o le \(\mathbf{E} = (12, -8, 28)\).
Fesili 4: Umi o le Vekita
Fuafua le umi (tele) o le vector \(\mathbf{A} = (1, 2, 2)\).
Talanoaga:
E mafai ona fuafuaina le umi po'o le tele o se vector \(\mathbf{A} = (A_x, A_y, A_z)\) e fa'aaoga ai le fua fa'atatau:
\[
|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
\]
Suitulaga i tau o loʻo tuʻuina atu:
\[
|\mathbf{A}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
O lea la, o le umi o le vector \(\mathbf{A}\) e 3.
Fesili 5: Oloa Fa'aputu
Afai e tu'uina atu ni vectors se lua, \(\mathbf{A} = (1, 0, -1)\) ma le \(\mathbf{B} = (2, 3, 4)\). Fa'atatau le dot product o nei vectors e lua.
Talanoaga:
O le fua fa'atatau o vectors e lua \(\mathbf{A} = (A_x, A_y, A_z)\) ma le \(\mathbf{B} = (B_x, B_y, B_z)\) e faia e ala i le fa'ateleina o vaega tutusa ona fa'aopoopoina lea. O lea la, ua ia i tatou:
\[
\mathbf{A} \cdot \mathbf{B} = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z
\]
Suitulaga i tau o loʻo tuʻuina atu:
\[
\mathbf{A} \cdot \mathbf{B} = (1 \cdot 2) + (0 \cdot 3) + (-1 \cdot 4) = 2 + 0 – 4 = -2
\]
O lea la, o le dot product o vectors \(\mathbf{A}\) ma le \(\mathbf{B}\) e -2.
Fesili 6: Oloa Fa'atasi
Afai e tu'uina atu ni vectors se lua, \(\mathbf{A} = (1, 2, 3)\) ma le \(\mathbf{B} = (4, 5, 6)\). Fa'atatau le fa'asologa o le oloa o nei vectors e lua.
Talanoaga:
O le fa'asologa o le oloa o vectors e lua \(\mathbf{A} = (A_x, A_y, A_z)\) ma le \(\mathbf{B} = (B_x, B_y, B_z)\) e faia e fa'aaoga ai le fua fa'atatau lenei:
\[
\mathbf{A} \times \mathbf{B} = \left( (A_y \cdot B_z – A_z \cdot B_y), (A_z \cdot B_x – A_x \cdot B_z), (A_x \cdot B_y – A_y \cdot B_x) \right)
\]
Suitulaga i tau o loʻo tuʻuina atu:
\[
\mathbf{A} \times \mathbf{B} = \left( (2 \cdot 6 – 3 \cdot 5), (3 \cdot 4 – 1 \cdot 6), (1 \cdot 5 – 2 \cdot 4) \right) = (12 – 15, 12 – 6, 5 – 8) = (-3, 6, -3)
\]
O lea la, o le fua fa'atatau o vectors \(\mathbf{A}\) ma le \(\mathbf{B}\) o le \(\mathbf{A} \times \mathbf{B} = (-3, 6, -3)\).
I'uga
O vectors e tolu-vaega i le Cartesian coordinate system o ni meafaigaluega taua i vaega eseese o le saienisi ma le inisinia. E ala i faʻataʻitaʻiga ma talanoaga o loʻo i luga, ua tatou vaʻaia ai le faʻatinoina o galuega faʻavae eseese i luga o vectors, e pei o le faʻaopoopoga, toʻesega, faʻatelega scalar, ma dot and cross products. O se malamalamaga mautu i nei manatu o le a matua aoga lava e le gata i le matematika ae faʻapea foi i faʻaoga aoga i le fisiki, inisinia, ma le saienisi komepiuta.
