Cov Lus Nug Piv Txwv Sib Tham Txog Cov Lus Siv, Cov Cim, thiab Cov Hom Vectors
Cov cim vector thiab nws txoj kev nkag siab yog qhov tseem ceeb hauv ntau ceg ntawm kev tshawb fawb, tshwj xeeb tshaj yog physics thiab lej. Kev siv cov vectors kom raug tuaj yeem pab txheeb xyuas cov teeb meem thiab nrhiav kev daws teeb meem zoo. Tsab xov xwm no tham txog cov lus thiab cov cim cuam tshuam nrog vectors, piav qhia lawv nrog cov piv txwv thiab cov lus piav qhia ntxaws ntxaws.
Cov Lus Txhais Vector
Yuav kom nkag siab txog cov vectors, peb yuav tsum xub nkag siab txog cov lus yooj yim:
1. Vector: Ib qho ntau uas muaj qhov loj (tus nqi loj) thiab kev coj. Cov vectors feem ntau yog cim los ntawm cov ntawv tuab xws li A, a, lossis los ntawm lub cim xub saum toj ntawm lawv xws li \(\vec{A}\).
2. Qhov Loj (Tus Nqi Loj): Qhov no yog qhov ntev lossis qhov loj ntawm lub vector. Nws yog qhia los ntawm | A | lossis \(\|\vec{A}\|\).
3. Taub Hau thiab Tail: Hauv kev sawv cev duab, cov vectors raug piav qhia ua xub. Qhov chaw pib ntawm xub hu ua tus tw thiab qhov chaw xaus ntawm xub hu ua lub taub hau.
4. Cov Vectors Sib Npaug: Cov Vectors uas sib luag rau ib leeg lossis nyob rau tib txoj kab kev ua haujlwm.
5. Collinear Vectors: Cov vectors uas nyob rau ntawm ib txoj kab ncaj.
6. Cov Vector Uas Tau Txais: Ib qho vector uas muaj tib qho txiaj ntsig zoo li qhov txiaj ntsig ua ke ntawm ob lossis ntau dua vectors.
Cov cim Vector
Cov cim vector muaj ntau txoj cai uas yuav tsum nkag siab kom txhais thiab sau cov vectors kom raug.
1. Cov Ntawv Loj thiab Cov Lus Cim Xub: Cov vectors feem ntau yog cim nrog cov ntawv loj lossis cov xub. Piv txwv li: A, B, lossis \(\vec{A}\).
2. Cov Vector Coordinates: Cov Vectors nyob rau hauv ob-seem (2D) qhov chaw yog cim raws li \(\vec{A} = (A_x, A_y)\), thaum nyob rau hauv peb-seem (3D) qhov chaw lawv yog cim raws li \(\vec{A} = (A_x, A_y, A_z)\).
3. Cov Vectors Hauv Paus: Hauv qhov chaw 2D thiab 3D, cov vectors hauv paus uas siv ntau yog \(\vec{i}\), \(\vec{j}\), thiab \(\vec{k}\), uas xa mus rau x, y, thiab z cov lus qhia, raws li.
4. Kev Ua Haujlwm Vector:
- Ntxiv: \(\vec{A} + \vec{B}\)
- Kev rho tawm: \(\vec{A} - \vec{B}\)
- Kev Sib Npaug Scalar: \(k\vec{A}\)
- Dot Multiplication (dot product): \(\vec{A} \cdot \vec{B}\)
- Kev Sib Npaug Sib Npaug (kev sib npaug sib npaug): \(\vec{A} \times \vec{B}\)
Cov Hom Vector
Muaj ntau hom vectors sib txawv nyob ntawm seb qhov xwm txheej thiab lawv cov yam ntxwv li cas:
1. Vector xoom: Ib qho vector uas muaj qhov loj ntawm 0 thiab tsis muaj kev taw qhia. Nws yog cim los ntawm 0 lossis \(\vec{0}\).
2. Chav Vector: Ib qho vector uas muaj qhov loj ntawm 1. Feem ntau siv los qhia kev taw qhia.
3. Txoj Haujlwm Vector: Ib qho vector uas qhia qhov chaw ntawm ib qho taw tes piv rau qhov keeb kwm (0,0,0).
4. Cov Vectors Sib Npaug thiab Cov Vectors Sib Npaug: Cov Vectors uas nyob rau tib qho kev taw qhia thiab kev taw qhia sib txawv, tab sis nyob rau ntawm tib txoj kab kev ua haujlwm.
5. Coplanar Vectors: Cov vectors uas nyob hauv tib lub dav hlau.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Lo lus nug 1: Xam qhov Vector Magnitude
Qhov loj ntawm lub vector \(\vec{A} = (3, 4)\) yog dab tsi?
Lus teb:
Yuav kom xam qhov loj ntawm lub vector \(\vec{A}\), peb siv cov mis:
\[\|\vec{A}\| = \sqrt{A_x^2 + A_y^2}\]
Hloov cov nqi rau hauv cov mis:
\[\|\vec{A}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
Yog li, qhov loj ntawm lub vector {\(\vec{A}\) yog 5.
Lo Lus Nug 2: Kev Ntxiv thiab Kev Rho Tawm ntawm Cov Vectors
Muab ob lub vectors \(\vec{A} = (2, 3)\) thiab \(\vec{B} = (1, -1)\). Xam \(\vec{A} + \vec{B}\) thiab \(\vec{A} – \vec{B}\).
Lus teb:
Ntxiv cov vectors \(\vec{A}\) thiab \(\vec{B}\):
\[\vec{A} + \vec{B} = (2, 3) + (1, -1) = (2 + 1, 3 - 1) = (3, 2)\]
Kev rho tawm ntawm cov vectors \(\vec{A}\) thiab \(\vec{B}\):
\[\vec{A} – \vec{B} = (2, 3) – (1, -1) = (2 – 1, 3 – (-1)) = (1, 4)\]
So, \(\vec{A} + \vec{B} = (3, 2)\) and \(\vec{A} – \vec{B} = (1, 4)\).
Lo lus nug 3: Cov khoom lag luam
Xam cov dot product ntawm ob lub vectors \(\vec{A} = (2, 3)\) thiab \(\vec{B} = (1, 4)\).
Lus teb:
Cov dot product ntawm ob lub vectors yog:
\[\vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y\]
Kev hloov tus nqi:
\[\vec{A} \cdot \vec{B} = 2 \cdot 1 + 3 \cdot 4 = 2 + 12 = 14\]
Yog li, qhov dot product ntawm \(\vec{A}\) thiab \(\vec{B}\) yog 14.
Lo lus nug 4: Cov khoom sib tshuam
Muab ob lub vectors hauv qhov chaw peb-seem \(\vec{A} = (1, 2, 3)\) thiab \(\vec{B} = (4, 5, 6)\). Xam qhov sib txawv ntawm cov khoom \(\vec{A} \times \vec{B}\).
Lus teb:
Qhov khoom sib tshuam ntawm ob lub vectors hauv qhov chaw peb-seem yog txhais tias yog tus txiav txim siab ntawm cov matrix hauv qab no:
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{vmatrix}
\]
Rau cov vectors \(\vec{A}\) thiab \(\vec{B}\):
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
\]
Xam raws li:
\[
\vec{A} \times \vec{B} = \vec{i}(2 \cdot 6 – 3 \cdot 5) – \vec{j}(1 \cdot 6 – 3 \cdot 4) + \vec{k}(1 \cdot 5 – 2 \cdot 4)
\]
\[
= \vec{i}(12 - 15) - \vec{j}(6 - 12) + \vec{k}(5 - 8)
\]
\[
= \vec{i}(-3) - \vec{j}(-6) + \vec{k}(-3)
\]
\[
= -3\vec{i} + 6\vec{j} – 3\vec{k}
\]
Yog li, qhov sib tshuam ntawm \(\vec{A}\) thiab \(\vec{B}\) yog \(\vec{A} \times \vec{B} = (-3, 6, -3)\).
Thaum daws teeb meem vector, kev nkag siab txog cov ntsiab lus yooj yim thiab cov lus siv yog qhov pib tseem ceeb. Tsab xov xwm no lub hom phiaj yog los muab kev nkag siab rau cov neeg nyeem txog ntau yam kev ua haujlwm vector thiab lawv cov hom sib txawv, uas yuav muaj txiaj ntsig zoo rau kev tshuaj xyuas lej thiab lub cev.
