Cov vectors yog ib lub tswv yim tseem ceeb hauv physics, siv los sawv cev rau qhov ntau thiab tsawg nrog ob qho tib si qhov loj thiab kev coj. Hauv physics, vectors feem ntau siv los piav qhia txog ntau yam xwm txheej xws li lub zog, qhov ceev, kev ua kom nrawm, thiab ntau ntxiv. Tsab xov xwm no yuav tham txog ntau qhov piv txwv ntawm cov teeb meem vector physics, nrog rau lawv cov kev daws teeb meem thiab cov lus piav qhia.
1. Kev Ntxiv thiab Rho Vector
Piv txwv lus nug 1:
Ob lub vectors \(\mathbf{A}\) thiab \(\mathbf{B}\) raug muab raws li nram no:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Xam:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Kev daws teeb meem:
Yuav ntxiv ob lub vectors, peb ntxiv lawv cov khoom sib cais.
1. \(\mathbf{A} + \mathbf{B}\):
\[
\mathbf{A} + \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) + (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – 2)\mathbf{i} + (4 + 5)\mathbf{j}
\]
\[
= 1\mathbf{i} + 9\mathbf{j}
\]
\[
\mathbf{A} + \mathbf{B} = \mathbf{i} + 9\mathbf{j}
\]
2. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Yog li, qhov tshwm sim yog:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Kev Sib Npaug Scalar (Dot Product)
Piv txwv lus nug 2:
Ob lub vectors \(\mathbf{C}\) thiab \(\mathbf{D}\) raug muab raws li nram no:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Xam cov khoom scalar (dot product) ntawm \(\mathbf{C}\) thiab \(\mathbf{D}\).
Kev daws teeb meem:
Qhov scalar product ntawm ob lub vectors \(\mathbf{C}\) thiab \(\mathbf{D}\) yog:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 x 3 + 2 x 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Yog li, qhov tshwm sim ntawm cov khoom lag luam scalar ntawm \(\mathbf{C}\) thiab \(\mathbf{D}\) yog 26.
3. Khoom Sib Txawv
Piv txwv lus nug 3:
Ob lub vectors \(\mathbf{E}\) thiab \(\mathbf{F}\) raug muab raws li nram no:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Xam qhov sib txawv ntawm \(\mathbf{E}\) thiab \(\mathbf{F}\).
Kev daws teeb meem:
Qhov khoom sib tshuam ntawm ob lub vectors \(\mathbf{E}\) thiab \(\mathbf{F}\) tuaj yeem suav los ntawm kev siv tus lej matrix determinant:
\[
\mathbf{E} \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
\]
Xam tus determinant ntawm lub matrix:
\[
\mathbf{E} \times \mathbf{F} = \mathbf{i} (2 \cdot 6 - 3 \cdot 5) - \mathbf{j} (1 \cdot 6 - 3 \cdot 4) + \mathbf{k} (1 \cdot 5 - 2 \cdot 4)
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
Yog li, qhov tshwm sim ntawm cov khoom sib tshuam ntawm \(\mathbf{E}\) thiab \(\mathbf{F}\) yog:
\[
\mathbf{E} \times \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Qhov loj ntawm Vector
Piv txwv lus nug 4:
Muab lub vector \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Xam qhov loj (ntev) ntawm lub vector \(\mathbf{G}\).
Kev daws teeb meem:
Qhov loj ntawm lub vector \(\mathbf{G}\) tuaj yeem suav los ntawm kev siv cov mis:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= 9 + 16
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
Yog li, qhov loj ntawm lub vector \(\mathbf{G}\) yog 5.
5. Kev daws teeb meem ntawm vector
Piv txwv lus nug 5:
Tus vector \(\mathbf{H}\) muaj qhov loj ntawm 10 units thiab tsim lub kaum sab xis ntawm 30° nrog rau x-axis. Txheeb xyuas cov khoom ntawm vector \(\mathbf{H}\) ntawm x- thiab y-axes.
Kev daws teeb meem:
Cov khoom ntawm vector \(\mathbf{H}\) ntawm x (\(\mathbf{H}_x\)) thiab y (\(\mathbf{H}_y\)) axes tuaj yeem suav los ntawm kev siv trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \sin(\theta)
\]
Nrog \(|\mathbf{H}| = 10\) thiab \(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \sin(30°)
\]
Cov nqi ntawm \(\cos(30°) = \frac{\sqrt{3}}{2}\) thiab \(\sin(30°) = \frac{1}{2}):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
Yog li, cov khoom ntawm vector \(\mathbf{H}\) yog:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Xaus
Hauv tsab xov xwm no, peb tau tham txog ntau yam teeb meem piv txwv uas cuam tshuam nrog vectors hauv physics, xws li vector ntxiv thiab rho tawm, scalar thiab cross multiplication, mus rau vector magnitude thiab kev daws teeb meem. Kev nkag siab txog lub tswv yim thiab kev ua haujlwm ntawm vectors yog qhov tseem ceeb hauv physics vim tias ntau yam xwm txheej ntuj tsim tuaj yeem piav qhia siv vectors. Vam tias, cov teeb meem piv txwv no yuav pab koj nkag siab lub tswv yim ntawm vectors tob dua.
