Nā ʻĀpana Vector: Nā Kumu, Nā Wehewehena, a me nā Hoʻohana
He manaʻo nui nā vectors i ka makemakika, ka physics, a me ka ʻenekinia. Ma nā ʻano ʻepekema like ʻole, hoʻohana pinepine ʻia lākou e wehewehe i nā nui i loaʻa ka nui a me ke kuhikuhi. Ma kēia ʻatikala, e ʻimi mākou i nā ʻāpana o kahi vector: e wehewehe ana i ke ʻano o kahi vector, pehea e hoʻokaʻawale ai i kahi vector i loko o kona mau ʻāpana, a e ʻimi ana i nā noi like ʻole a me nā hopena o nā vectors i ke ola o kēlā me kēia lā a me ka ʻepekema.
Ke Hoʻomaopopo nei i nā Vectors
ʻO ka vector kahi nui i loaʻa ʻole kahi waiwai (magnitude) akā he kuhikuhi pū kekahi. ʻAʻole e like me nā scalars, nona wale nō kahi waiwai (e like me ka mahana a i ʻole ka nuipa), loaʻa i nā vectors kēia mau ʻano nui ʻelua a hoʻohana ʻia e hōʻike i nā hanana kahi i lilo ai ke kuhikuhi i mea nui, e like me ka wikiwiki, ka ikaika, a me ka neʻe ʻana.
Ma ke ʻano makemakika, hiki ke hōʻike ʻia kahi vector ma kahi ʻelua-dimensional (2D) e like me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), kahi ʻo \(v_x\) a me \(v_y\) nā ʻāpana o ka vector \(\mathbf{v}\) ma nā kuhikuhi x- a me y. Ma kahi ʻekolu-dimensional (3D), hiki ke hōʻike ʻia kahi vector e like me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\).
Hōʻike Vector a me nā ʻĀpana
No ka hoʻomaopopo ʻana i ke kumumanaʻo o nā ʻāpana vector, pono mākou e ʻike e hiki ke hoʻokaʻawale ʻia nā vectors i nā ʻāpana e pili ana i kēlā me kēia axis coordinate. No ka laʻana, ma kahi ākea ʻelua-dimensional, hiki ke hoʻokaʻawale ʻia kahi vector \(\mathbf{v}\) i ʻelua mau ʻāpana: \(v_x\) (ka ʻāpana ma ke kuhikuhi-x) a me \(v_y\) (ka ʻāpana ma ke kuhikuhi-y).
Ma ke ʻano geometric, inā mākou e kaha kiʻi i kahi vector ma ka mokulele hoʻonohonoho Cartesian, hiki ke hoʻohālikelike ʻia me kahi pua e kuhikuhi ana mai ke kumu \((0,0)\) a i ke kiko \((v_x, v_y)\). Hiki ke ʻike ʻia nā ʻāpana \(v_x\) a me \(v_y\) ma ke ʻano he lōʻihi o nā projections o ka vector ma luna o nā axes x- a me y.
Ma kahi ākea ʻekolu-dimensional, hiki ke hoʻokaʻawale ʻia kahi vector i ʻekolu mau ʻāpana: \(v_x\) (ka ʻāpana kuhikuhi-x), \(v_y\) (ka ʻāpana kuhikuhi-y), a me \(v_z\) (ka ʻāpana kuhikuhi-z). I nā huaʻōlelo ʻē aʻe, hiki ke hōʻike ʻia kahi vector ma kahi ākea ʻekolu-dimensional e kahi pua e kuhikuhi ana mai ke kumu \((0,0,0)\) a i ke kiko \((v_x, v_y, v_z)\).
Ka nui a me ke kuhikuhi o nā Vectors
No ka helu ʻana i ka nui a i ʻole ka lōʻihi o kahi vector \(\mathbf{v}\), hoʻohana mākou i ke ʻano hana:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
\]
no ka wahi ʻelua-dimensional, a:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
no ka hakahaka ʻekolu-dimensional. Ua kapa pinepine ʻia kēia nui vector ʻo kona nui a hōʻike i ka lōʻihi o ka vector.
Hiki ke hōʻike ʻia ke kuhikuhi o kahi vector ma ke ʻano o kona kihi e pili ana i nā axis coordinate. Ma kahi ākea ʻelua-dimensional, hiki ke helu ʻia ke kuhikuhi o kahi vector \(\mathbf{v}\) e hana ana i kahi kihi \(\theta\) me ka axis-x me ka hoʻohana ʻana i ka trigonometry:
\[
\theta = \tan^{-1}\left(\frac{v_y}{v_x}\ʻākau)
\]
Ma kahi ākea ʻekolu-dimensional, ʻoi aku ka paʻakikī o ka hoʻoholo ʻana i ke kuhikuhi, no ka mea, pono mākou e helu i nā kihi me kēlā me kēia axis coordinate. ʻO ka maʻamau, hoʻohana ʻia kahi ʻōnaehana poepoe e hōʻike i ke kuhikuhi ma kahi ākea ʻekolu-dimensional.
Nā Hana ma nā Vectors
Hoʻohui a me ka Hoʻemi
Hoʻohui ʻia nā vectors ʻelua ma ka hoʻohui ʻana i nā ʻāpana pākahi o nā vectors ʻelua. No ka laʻana, inā \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \end{bmatrix}\) a me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), a laila:
\[
\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_x + v_x \\ u_y + v_y \end{bmatrix}
\]
Ua helu ʻia ka unuhi vector ma ke ʻano like:
\[
\mathbf{u} – \mathbf{v} = \begin{bmatrix} u_x – v_x \\ u_y – v_y \end{bmatrix}
\]
Hoʻonui Scala
ʻO ka hoʻonui ʻana i kahi vector me kahi scalar (he helu hoʻokahi) e hana ʻia ma ka hoʻonui ʻana i kēlā me kēia ʻāpana o ka vector me ka scalar. No ka laʻana, inā he scalar ʻo \(k\) a ʻo \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), a laila:
\[
k \cdot \mathbf{v} = \begin{bmatrix} k \cdot v_x \\ k \cdot v_y \end{bmatrix}
\]
Hoʻonui ʻana i ke kiko a me ke keʻa
I loko o ka lewa ʻekolu-dimensional, aia ʻelua ʻano o ka hoʻonui vector: ka hoʻonui kiko a me ka hoʻonui keʻa.
1. Hoʻonui ʻana i nā kiko:
Ua wehewehe ʻia ka huahana kiko o nā vectors ʻelua \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix}\) a me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\) penei:
\[
\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z
\]
ʻO ka hopena o kahi huahana kiko he scalar. Hoʻohana pinepine ʻia ka huahana kiko e hoʻoholo i ka nui o nā vectors ʻelua e like a orthogonal paha kekahi i kekahi.
2. Hoʻonui Keʻa:
ʻO ka huahana kea o ʻelua mau vectors i loko o kahi ākea ʻekolu-dimensional e hana i kahi vector hou e kū pololei ana i nā vectors mua ʻelua. Inā \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix}\) a me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\), a laila ua wehewehe ʻia ka huahana kea penei:
\[
\mathbf{u} \manawa \mathbf{v} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_x & u_y & u_z \\
v_x & v_y & v_z
\end{vmatrix}
\]
Hoʻonohonoho Vector
ʻO ke ʻano maʻamau ke kaʻina hana o ka hoʻololi ʻana i kahi vector i loko o kahi vector unit (kahi vector o ka lōʻihi 1) me ke kuhikuhi like. Loaʻa ka vector unit \(\mathbf{\hat{v}}\) o \(\mathbf{v}\) ma ka puʻunaue ʻana i kēlā me kēia o kona mau ʻāpana ma ka lōʻihi (ka nui) o ka vector:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}
\]
Nā Hoʻohana ʻana o nā Vectors i ke Ola o kēlā me kēia lā a me ka ʻEpekema
He nui nā hoʻohana o nā Vectors i ke ola o kēlā me kēia lā a me ka ʻepekema. Eia kekahi mau laʻana:
1. ʻIke kino:
I ke ʻano physics, hoʻohana ʻia nā vectors e wehewehe i nā nui like ʻole e like me ka wikiwiki, ka wikiwiki, ka ikaika, a me ka momentum. No ka laʻana, hiki ke kālailai ʻia ka neʻe ʻana o kahi mea me ka hoʻohana ʻana i nā vectors wikiwiki a me ka wikiwiki.
2. ʻAno hana:
I loko o ka ʻenekinia, hoʻohana ʻia nā vectors no ka nānā ʻana i ke ʻano, ka hoʻolālā mīkini, a me nā noi ʻenekinia like ʻole. No ka laʻana, ʻo ka nānā ʻana i ke koʻikoʻi a me ke kaumaha i loko o kahi mea e pili pinepine ana i ka hoʻohana ʻana i nā vectors.
3. Nā Kiʻi Kamepiula:
Hoʻohana ʻia nā vectors i nā kiʻi kamepiula e wehewehe i ke kūlana, ke kuhikuhi ʻana, a me ka neʻe ʻana o nā mea. I ka papahana kiʻi, hoʻohana ʻia nā vectors no nā hoʻololi e like me ka unuhi ʻana, ka hoʻohuli ʻana, a me ka scaling.
4. Hoʻokele:
Hoʻohana ʻia nā vectors i nā ʻōnaehana hoʻokele e hoʻoholo i ke kuhikuhi a me ka mamao ma waena o ʻelua mau wahi. Hoʻohana ʻo GPS a me nā ʻōnaehana hoʻokele ʻē aʻe i nā vectors e helu i nā ala a kuhikuhi i nā mea hoʻohana.
5. Hoʻokele waiwai:
I ka hoʻokele waiwai, hiki ke hoʻohana ʻia nā vectors e wehewehe i nā makemake o nā mea kūʻai aku a i ʻole nā waihona hoʻopukapuka kālā. Hoʻokomo pinepine ka loiloi ʻikepili multivariable i ka hoʻohana ʻana i nā vectors.
Ka hopena
He manaʻo koʻikoʻi a maʻalahi hoʻi nā Vectors i ka makemakika a me nā ʻoihana ʻepekema ʻē aʻe. Ma ka hoʻomaopopo ʻana i nā ʻāpana o nā vectors a me nā hana like ʻole e hiki ke hana ʻia ma luna o lākou, hiki iā mākou ke hoʻopili i kēia manaʻo e hoʻoponopono i nā pilikia hana a me nā pilikia ʻepekema. Me kā lākou hōʻike makemakika ikaika, hāʻawi nā vectors i kahi mea hana kūpono no ka wehewehe ʻana a me ka nānā ʻana i nā hanana like ʻole e pili ana i nā nui me nā kuhikuhi.
