Mavector ipfungwa inokosha mufizikisi, inoshandiswa kumiririra huwandu hune hukuru uye gwara. Mufizikisi, mavector anowanzo shandiswa kutsanangura zviitiko zvakasiyana-siyana zvakaita sesimba, kumhanya, kukurumidza, nezvimwewo. Chinyorwa chino chichakurukura mienzaniso yakati wandei yezvinetso zvevector zvefizikisi, pamwe chete nemhinduro dzazvo netsananguro.
1. Kuwedzera nekubvisa mavekita
Muenzaniso Mubvunzo 1:
Mavector maviri \(\mathbf{A}\) uye \(\mathbf{B}\) anopiwa seizvi:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Verenga:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Mhinduro:
Kuti tiwedzere mavector maviri, tinowedzera zvikamu zvawo zvakasiyana.
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. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Saka, mhedzisiro yacho ndeiyi:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Kuwanda kweScalar (Chigadzirwa cheDot)
Muenzaniso Mubvunzo 2:
Mavector maviri \(\mathbf{C}\) uye \(\mathbf{D}\) anopiwa seizvi:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Verenga chigadzirwa chescalar (dot product) che \(\mathbf{C}\) uye \(\mathbf{D}\).
Mhinduro:
Chigadzirwa che scalar chema vector maviri \(\mathbf{C}\) uye \(\mathbf{D}\) ndeichi:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Saka, mhedzisiro yechibereko che scalar che \(\mathbf{C}\) uye \(\mathbf{D}\) i26.
3. Chigadzirwa Chinosiyana
Muenzaniso Mubvunzo 3:
Mavector maviri \(\mathbf{E}\) uye \(\mathbf{F}\) anopiwa seizvi:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Verengai mhedzisiro ye \(\mathbf{E}\) uye \(\mathbf{F}\).
Mhinduro:
Chibereko chekubatanidza mavector maviri \(\mathbf{E}\) uye \(\mathbf{F}\) chinogona kuverengerwa uchishandisa matrix determinant:
\[
\mathbf{E} \nguva \mathbf{F} = \kutanga{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4 ne5 & 6
\kuguma{vmatrix}
\]
Verenga chinongedzo che matrix:
\[
\mathbf{E} \nguva \mathbf{F} = \mathbf{i} (2 \cdot 6 - 3 \cdot 5) - \mathbf{j} (1 \cdot 6 - 3 \cdot 4) + \mathbf{k} (1 \cdot 5 - 4 \cdot)
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
Saka, mhedzisiro yemusanganiswa we \(\mathbf{E}\) uye \(\mathbf{F}\) ndeiyi:
\[
\mathbf{E} \nguva \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Kukura kweVekita
Muenzaniso Mubvunzo 4:
Zvichienderana nevector \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Verenga hukuru (hurefu) hwevector \(\mathbf{G}\).
Mhinduro:
Hukuru hwevector \(\mathbf{G}\) hunogona kuverengerwa uchishandisa fomura:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
Saka, hukuru hwevector \(\mathbf{G}\) ndi5.
5. Kugadziriswa kweVector
Muenzaniso Mubvunzo 5:
Vector \(\mathbf{H}\) ine hukuru hwemayuniti gumi uye inoumba kona ye30° ne x-axis. Sarudza zvikamu zvevector \(\mathbf{H}\) pa x- uye y-axes.
Mhinduro:
Zvikamu zvevector \(\mathbf{H}\) pa x (\(\mathbf{H}_x\)) uye y (\(\mathbf{H}_y\)) zvinogona kuverengerwa uchishandisa trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \chivi(\theta)
\]
Ne \(|\mathbf{H}| = 10\) uye \(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \chivi(30°)
\]
Kukosha kwe \(\cos(30°) = \frac{\sqrt{3}}{2}\) uye \(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
Saka, zvikamu zvevector \(\mathbf{H}\) ndeizvi:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Mhedziso
Muchinyorwa chino, takurukura nezvematambudziko akati wandei ane chekuita nemavector mufizikisi, kubva pakuwedzera nekubvisa mavector, scalar necross multiplication, kusvika pakukura kwevector uye resolution. Kunzwisisa pfungwa nekushanda kwemavector kwakakosha mufizikisi, sezvo zviitiko zvakawanda zvechisikigo zvinogona kutsanangurwa uchishandisa mavector. Tinovimba kuti matambudziko aya emuenzaniso achakubatsira kunzwisisa pfungwa yemavector zvakadzama.
