Cov Lus Txhais thiab Hom ntawm Vector Notation
Thaum tham txog lej, physics, thiab computer science, lub tswv yim ntawm vectors feem ntau yog ib qho tseem ceeb uas yuav tsum nkag siab. Vectors tsis yog tsuas yog cov tswv yim abstract xwb; lawv muaj feem cuam tshuam rau ntau yam xwm txheej, xws li kev tshuaj xyuas cov ntaub ntawv, computer graphics, thiab physics simulations. Hauv tsab xov xwm no, peb yuav tham txog vector terminology thiab notation, tom qab ntawd tshawb nrhiav ntau hom vectors pom muaj nyob rau hauv cov kev qhuab qhia no.
Cov Lus Txhais thiab Cov Cim Vector
1. Cov Vectors thiab Scalars
Ib lub vector yog ib qho lej uas muaj ob qho loj thiab kev coj. Qhov sib txawv, ib lub scalar yog ib qho nqi uas muaj tsuas yog qhov loj thiab tsis muaj kev coj. Piv txwv li, qhov ceev ntawm 5 m / s yam tsis qhia qhov kev coj yog ib lub scalar, thaum qhov ceev ntawm 5 m / s sab hnub tuaj yog ib lub vector.
2. Cov cim vector
Feem ntau cov vectors yog cim los ntawm tsab ntawv me me xws li v, lossis los ntawm tus xub saum tsab ntawv xws li \(\vec{v}\). Piv txwv li, yog tias peb muaj vector v uas nws cov ntsiab lus yog \(v_1, v_2, v_3\), ces qhov no tuaj yeem sau ua:
\[ \vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \]
Lwm txoj hauv kev los sau cov vectors, tshwj xeeb tshaj yog nyob rau hauv ob lossis peb-seem, yog siv tus qauv. Piv txwv li:
\[ \vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k} \]
qhov twg \(\hat{i}, \hat{j}\), thiab \(\hat{k}\) yog cov vectors unit ntawm x, y, thiab z axes.
Hom Vectors
1. Txoj Haujlwm Vector
Ib qho vector txoj hauj lwm yog ib qho vector uas piav qhia txog qhov chaw ntawm ib qho chaw hauv qhov chaw piv rau ib qho chaw siv, feem ntau yog qhov chaw O (lub hauv paus chiv keeb). Yog tias qhov chaw P muaj cov kev sib koom ua ke (x, y, z) hauv qhov chaw 3D, ces qhov chaw vector \(\vec{r}\) tuaj yeem qhia tau li:
\[ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \]
2. Kev Hloov Chaw Vector
Ib lub vector hloov chaw piav qhia txog kev hloov pauv ntawm qhov chaw ntawm ib qho chaw ntawm ib qho chaw mus rau lwm qhov. Xav tias qhov chaw A muaj cov coordinates (x1, y1, z1) thiab qhov chaw B muaj cov coordinates (x2, y2, z2). Lub vector hloov chaw \(\vec{d}\) ntawm A mus rau B tuaj yeem sau ua:
\[ \vec{d} = (x2 – x1)\hat{i} + (y2 – y1)\hat{j} + (z2 – z1)\hat{k} \]
3. Qhov ceev Vector
Qhov ceev yog ib qho vector uas qhia txog qhov nrawm ntawm kev hloov pauv ntawm qhov chaw ntawm ib yam khoom ib lub sijhawm. Yog tias \(\vec{r}(t)\) yog ib qho kev ua haujlwm ntawm qhov chaw piv rau lub sijhawm, qhov ceev vector \(\vec{v}(t)\) yog qhov derivative ntawm \(\vec{r}(t)\) piv rau lub sijhawm t:
\[ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} \]
4. Kev nrawm Vector
Tus vector acceleration yog tus derivative ntawm tus vector velocity piv rau lub sijhawm. Nws qhia txog tus nqi ntawm kev hloov pauv ntawm tus velocity ntawm ib yam khoom ib lub sijhawm. Yog tias \(\vec{v}(t)\) yog ib qho function ntawm tus velocity piv rau lub sijhawm, tus vector acceleration \(\vec{a}(t)\) yog tus derivative ntawm \(\vec{v}(t)\):
\[ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} \]
5. Lub zog Vector
Raws li Newton txoj cai thib ob, lub zog yog cov khoom ntawm qhov hnyav thiab kev nrawm. Lub zog kuj yog ib qho vector vim nws muaj ob qho tib si qhov loj thiab kev coj. Yog tias m yog qhov hnyav thiab \(\vec{a}\) yog qhov vector kev nrawm, ces lub zog vector \(\vec{F}\) tuaj yeem qhia tau tias:
\[ \vec{F} = m\vec{a} \]
6. Chav Vector
Ib lub vector unit yog ib lub vector uas muaj qhov loj (ntev) ntawm ib qho. Lub vector unit ntawm ib lub vector \(\vec{v}\) tuaj yeem tau los ntawm kev faib \(\vec{v}\) los ntawm nws qhov loj. Yog tias \(\vec{v}\) muaj qhov loj ntawm \(||\vec{v}||\), ces nws lub vector unit tuaj yeem sau ua:
\[ \hat{v} = \frac{\vec{v}}{||\vec{v}||} \]
7. Vector xoom
Ib lub vector xoom yog ib lub vector uas nws cov khoom sib xyaw yog xoom, thiab feem ntau yog cim los ntawm \(\vec{0}\). Lub vector no tsis muaj kev taw qhia thiab nws qhov loj yog xoom. Ib qho piv txwv hauv qhov chaw peb-seem yog:
\[ \vec{0} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \]
8. Cov Vectors Orthogonal
Ob lub vectors raug hais tias yog orthogonal yog tias lawv cov khoom sab hauv yog xoom. Yog tias \(\vec{u}\) thiab \(\vec{v}\) yog ob lub vectors, ces lawv yog orthogonal yog tias:
\[ \vec{u} \cdot \vec{v} = 0 \]
9. Collinear Vectors thiab Coplanar Vectors
Ob lub vectors raug hu ua collinear yog tias lawv pw raws tib txoj kab ncaj lossis sib luag. Lawv tuaj yeem qhia ua scalar multiples ntawm ib leeg. Piv txwv li:
\[ \vec{v} = k\vec{u} \]
rau qee qhov scalar \(k\).
Lub caij no, peb lub vectors raug hais tias yog coplanar yog tias lawv pw hauv tib lub dav hlau. Lawv tuaj yeem qhia tau tias yog kev sib xyaw ua ke ntawm ob lub vectors.
Kev Ua Haujlwm ntawm Vectors
1. Kev Ntxiv thiab Rho Vector
Kev ntxiv vector yog ua los ntawm kev ntxiv lawv cov khoom sib xws. Yog tias \(\vec{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}\) thiab \(\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), ces:
\[ \vec{u} + \vec{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix} \]
Kev rho tawm yog ua los ntawm kev rho tawm cov khoom sib xws:
\[ \vec{u} – \vec{v} = \begin{pmatrix} u_1 – v_1 \\ u_2 – v_2 \\ u_3 – v_3 \end{pmatrix} \]
2. Kev Sib Npaug Scalar
Kev sib npaug ntawm cov lej (scalar multiplication) yog ib qho kev ua haujlwm uas muaj ib lub vector nrog ib lub lej (scalar) (tus nqi lej). Yog tias k yog ib lub lej thiab \(\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), ces:
\[ k\vec{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix} \]
3. Khoom Sab Hauv (Khoom Muag)
Cov khoom sab hauv ntawm ob lub vectors \(\vec{u}\) thiab \(\vec{v}\) yog ib qho scalar. Nws tuaj yeem suav los ntawm:
\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]
4. Khoom Sib Txawv
Qhov sib tshuam ntawm ob lub vectors \(\vec{u}\) thiab \(\vec{v}\) tsim ib lub vector tshiab uas yog orthogonal rau ob lub vectors no. Hauv qhov chaw peb-seem, qhov no yog xam raws li:
\[ \vec{u} \times \vec{v} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix} \]
Xaus
Kev nkag siab txog cov lus thiab cov cim qhia txog vector, nrog rau lawv cov hom, yog qhov tseem ceeb heev rau ntau yam kev kawm txog kev tshawb fawb. Vectors tsis yog tsuas yog cov lej xwb, tab sis kuj yog cov cuab yeej muaj zog hauv kev tshuaj xyuas physics, engineering, thiab kev siv tshuab xov xwm. Yog tias peb nkag siab zoo txog cov ntsiab lus tseem ceeb no, peb tuaj yeem daws cov teeb meem nyuaj hauv ntau qhov chaw.
