Nā Vectors Kolamu a me nā Vectors Lālani: Nā Kumu i ka Makemakika a me kā lākou Hoʻohana
I ka makemakika a me ka ʻepekema, he manaʻo nui ke kumumanaʻo o nā vectors. Hoʻohana ʻia nā vectors e hōʻike i nā nui me ke kuhikuhi a me ka nui. Ma waho aʻe o ko lākou hoʻohana ʻana i ka makemakika, loaʻa pū nā vectors i nā noi ma nā ʻano like ʻole e like me ke kino, ka ʻenekinia, a me nā kiʻi kamepiula. Ma ke ʻano o ka algebra linear, ua māhele pinepine ʻia nā vectors i ʻelua ʻano nui: nā vectors kolamu a me nā vectors lālani. E ʻimi hohonu kēia ʻatikala i nā manaʻo o nā vectors kolamu a me nā vectors lālani, a me kā lākou noi ma nā ʻano like ʻole.
Nā Wehewehena a me nā Hōʻailona
Vector Kolamu
ʻO ka vector kolamu kahi vector i hōʻike ʻia ma ke ʻano he kolamu kū pololei. ʻO ka hōʻailona laulā no ka vector kolamu penei:
\[
\mathbf{v} = \begin{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\end{bmatrix}
\]
Ma kahi o \(v_1, v_2, \ldots, v_n\) nā mea o ka vector. ʻO ka helu o nā mea i loko o ka vector e hōʻike ana i ka nui o ka vector.
Vector Laina
I ka hoʻohālikelike ʻana, ʻo ka vector lālani he vector i hōʻike ʻia ma ke ʻano he lālani ʻaoʻao. ʻO ka hōʻailona maʻamau no ka vector lālani penei:
\[
\mathbf{u} = \begin{bmatrix}
u_1 & u_2 & \cdots & u_n
\end{bmatrix}
\]
E like me ka vector kolamu, ʻo \(u_1, u_2, \ldots, u_n\) nā ʻāpana o ka vector me nā ana o ka vector.
Nā Hana Kumu me nā Vectors Kolamu a me nā Vectors Lālani
Hoʻohui a me ka Hoʻemi
Hiki ke hoʻohui a unuhi ʻia nā vector kolamu a me nā vector lālani inā like ko lākou ana. No ka laʻana, no nā vector kolamu ʻelua \(\mathbf{v}\) a me \(\mathbf{w}\) nona nā mea \(v_i\) a me \(w_i\), ʻo ka hoʻohui ʻana penei:
\[
\mathbf{v} + \mathbf{w} = \begin{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\end{bmatrix} + \begin{bmatrix}
w_1 \\
w_2 \\
\vdots \\
w_n
\end{bmatrix} = \begin{bmatrix}
v_1 + w_1 \\
v_2 + w_2 \\
\vdots \\
v_n + w_n
\end{bmatrix}
\]
No nā vectors lālani, ua like ke kumumanaʻo:
\[
\mathbf{u} + \mathbf{t} = \begin{bmatrix}
u_1 & u_2 & \cdots & u_n
\end{bmatrix} + \begin{bmatrix}
t_1 & t_2 & \cdots & t_n
\end{bmatrix} = \begin{bmatrix}
u_1 + t_1 & u_2 + t_2 & \cdots & u_n + t_n
\end{bmatrix}
\]
Hoʻonui Scala
ʻO ka hoʻonui scalar e pili ana i ka hoʻonui ʻana i kēlā me kēia element o kahi vector me kahi helu scalar. No ka laʻana, inā ʻo ka scalar \(c\) a me ka vector kolamu \(\mathbf{v}\), a laila:
\[
c\mathbf{v} = c \begin{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\end{bmatrix} = \begin{bmatrix}
cv_1 \\
cv_2 \\
\vdots \\
cv_n
\end{bmatrix}
\]
A inā ʻo ka vector lālani \(\mathbf{u}\):
\[
c\mathbf{u} = c \begin{bmatrix}
u_1 & u_2 & \cdots & u_n
\end{bmatrix} = \begin{bmatrix}
cu_1 & cu_2 & \cdots & cu_n
\end{bmatrix}
\]
Hoʻonui Vector
Hoʻokomo ka hoʻonui vector i kekahi mau ʻano mai ka huahana kiko a i ka huahana keʻa.
No nā vectors kolamu ʻelua \(\mathbf{v}\) a me \(\mathbf{w}\), ua hōʻike ʻia ka huahana kiko penei:
\[
\mathbf{v} \cdot \mathbf{w} = \sum_{i=1}^n v_i w_i
\]
ʻO ka hopena o ka huahana kiko he scalar. Eia nō naʻe, ua wehewehe wale ʻia ka huahana kea no nā vectors ma kahi ʻekolu-dimensional a hana i kahi vector hou i orthogonal i nā vectors mua ʻelua.
Nā noi ma nā kahua like ʻole
ʻO ke kinoea
I ke ʻano physics, hoʻohana pinepine ʻia nā vectors kolamu a me nā vectors lālani e hōʻike i nā nui kino like ʻole e like me ka wikiwiki, ka wikiwiki, a me nā kahua ikaika. No ka laʻana, hiki ke hōʻike ʻia ka wikiwiki o ka umekaumaha ma kahi kiko ma ka lewa ma ke ʻano he vector kolamu ʻekolu-dimensional:
\[
\mathbf{a} = \begin{bmatrix}
0 \\
-9.8 \\
0
\end{bmatrix} \, \text{m/s}^2
\]
ʻEnekinia a me ka ʻenehana
I loko o ka ʻenekinia, ʻoi aku hoʻi i ka loiloi kūkulu, hoʻohana pinepine ʻia nā vectors kolamu e hōʻike i nā ikaika a me nā manawa i loko o nā hale. No ka laʻana, hiki ke hōʻike ʻia nā ikaika ma nā wahi pili i loko o kahi hale kiʻi ma ke ʻano he mau vectors kolamu:
\[
\mathbf{F} = \begin{bmatrix}
F_x \\
F_y \\
F_z
\end{bmatrix}
\]
Ma kahi o \(F_x, F_y,\) a me \(F_z\) nā ʻāpana ikaika ma nā kuhikuhi orthogonal ʻekolu.
ʻEpekema Kamepiula a me nā Kiʻi Kamepiula
I ke kamepiula, he mea nui nā vectors no ka hōʻike ʻikepili a me ka hoʻoponopono ʻana. I nā kiʻi kamepiula, hoʻohana ʻia nā vectors e hōʻike i nā kiko, nā vectors kūlana, a me nā hoʻololi. No ka laʻana, hiki ke hōʻike ʻia kahi kiko i kahi ākea ʻekolu-dimensional ma ke ʻano he vector kolamu:
\[
\mathbf{p} = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
\]
Hōʻike ʻia hoʻi nā hoʻololi ʻana e like me nā unuhi, nā hoʻohuli, a me nā unahi me ka hoʻohana ʻana i nā matrices e hana ana ma nā vectors kolamu a lālani paha.
Ka Hoʻoponopono ʻana i nā ʻŌnaehana o nā Kaulike Linear
Hoʻohana pinepine ʻia nā vector kolamu a me nā vector lālani i ka hoʻoponopono ʻana i nā ʻōnaehana o nā hoʻohālikelike linear. No ka laʻana, ʻo kēia ka ʻōnaehana o nā hoʻohālikelike linear:
\[
nā hihia
a_{11}x_1 + a_{12}x_2 = b_1 \\
a_{21}x_1 + a_{22}x_2 = b_2
nā hihia
\]
Hiki ke hōʻike ʻia ma ke ʻano matrix penei:
\[
\begin{bmatrix}
he_{11} a me he_{12} \\
he_{21} a me he_{22}
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
b_1 \\
b_2
\end{bmatrix}
\]
ʻO kēia ʻano hana e maʻalahi loa ai ka hoʻohana ʻana i nā ʻano algebra linear e like me ka Gaussian elimination, LU decomposition, a i ʻole nā ʻano iterative no nā ʻōnaehana paʻakikī.
Ka hopena
ʻO nā vector kolamu a me nā vector lālani he mau mea nui e ʻike pinepine ʻia he maʻalahi akā he nui nā noi ma nā ʻano ʻepekema like ʻole a me ka ʻenekinia. ʻO ka hoʻomaopopo ʻana i nā kumu o nā hana vector he hana mua koʻikoʻi ia i ka hoʻopaʻa ʻana i ka algebra linear a me nā aʻo makemakika ʻē aʻe. Hāʻawi nā mea ʻelua i nā ala kūpono e hōʻike a hoʻoponopono i ka ʻikepili ma nā ʻano ʻano like ʻole, mai ka physics a me ka ʻenekinia a hiki i ka ʻepekema kamepiula. ʻO ka hoʻomaopopo hohonu ʻana i nā vector kolamu a me nā vector lālani hiki ke hoʻomākaukau i ke ala no nā manaʻo paʻakikī a me nā noi honua maoli.
