Maveketa ndi lingaliro lofunika kwambiri mu fizikisi, lomwe limagwiritsidwa ntchito kuyimira kuchuluka kwa zinthu ndi kukula kwake. Mu fizikisi, maveketa nthawi zambiri amagwiritsidwa ntchito pofotokoza zochitika zosiyanasiyana monga mphamvu, liwiro, kuthamanga, ndi zina zambiri. Nkhaniyi ikambirana zitsanzo zingapo za mavuto a veketa a fizikisi, pamodzi ndi mayankho ndi mafotokozedwe ake.
1. Kuwonjezera ndi Kuchotsa Vekitala
Chitsanzo cha Funso 1:
Ma vekitala awiri \(\mathbf{A}\) ndi \(\mathbf{B}\) aperekedwa motere:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Werengerani:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Yankho:
Kuti tiwonjezere ma vector awiri, timawonjezera zigawo zawo padera.
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. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Chifukwa chake, zotsatira zake ndi izi:
\[
\mathbf{A} - \mathbf{B} = 5\mathbf{i} - \mathbf{j}
\]
2. Kuchulukitsa kwa Scalar (Dot Product)
Chitsanzo cha Funso 2:
Ma vekitala awiri \(\mathbf{C}\) ndi \(\mathbf{D}\) aperekedwa motere:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Werengerani chinthu cha scalar (dot product) cha \(\mathbf{C}\) ndi \(\mathbf{D}\).
Yankho:
Chogulitsa cha scalar cha ma vector awiri \(\mathbf{C}\) ndi \(\mathbf{D}\) ndi:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Kotero, zotsatira za zotsatira za scalar za \(\mathbf{C}\) ndi \(\mathbf{D}\) ndi 26.
3. Zogulitsa Zosiyanasiyana
Chitsanzo cha Funso 3:
Ma vekitala awiri \(\mathbf{E}\) ndi \(\mathbf{F}\) aperekedwa motere:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Werengerani zotsatira zosakaniza za \(\mathbf{E}\) ndi \(\mathbf{F}\).
Yankho:
Chogulitsa chophatikizana cha ma vector awiri \(\mathbf{E}\) ndi \(\mathbf{F}\) chikhoza kuwerengedwa pogwiritsa ntchito matrix determinant:
\[
\mathbf{E} \nthawi \mathbf{F} = \kuyamba{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
\]
Werengerani chizindikiro cha matrix:
\[
\mathbf{E} \nthawi \mathbf{F} = \mathbf{i} (2 \cdot 6 - 3 \cdot 5) - \mathbf{j} (1 \cdot 6 - 3 \cdot 4) + \mathbf{k} (1 \cdot 5 - 4) \cdot
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
Kotero, zotsatira za zotsatira zosakanikirana za \(\mathbf{E}\) ndi \(\mathbf{F}\) ndi:
\[
\mathbf{E} \nthawi \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Kukula kwa Vekitala
Chitsanzo cha Funso 4:
Popeza vekitala \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Werengani kukula (kutalika) kwa vekitala \(\mathbf{G}\).
Yankho:
Kukula kwa vekitala \(\mathbf{G}\) kungawerengedwe pogwiritsa ntchito fomula iyi:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
Kotero, kukula kwa vekitala \(\mathbf{G}\) ndi 5.
5. Kusasinthika kwa Vekitala
Chitsanzo cha Funso 5:
Vekitala \(\mathbf{H}\) ili ndi kukula kwa mayunitsi 10 ndipo imapanga ngodya ya 30° ndi x-axis. Dziwani zigawo za vekitala \(\mathbf{H}\) pa ma axes a x- ndi y.
Yankho:
Zigawo za vekitala \(\mathbf{H}\) pa ma axes a x (\(\mathbf{H}_x\)) ndi y (\(\mathbf{H}_y\)) zitha kuwerengedwa pogwiritsa ntchito trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \kos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \ tchimo (\ theta)
\]
Ndi \(|\mathbf{H}| = 10\) ndi \(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \sin(30°)
\]
Ma values a \(\cos(30°) = \frac{\sqrt{3}}{2}\) ndi \(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
Kotero, zigawo za vekitala \(\mathbf{H}\) ndi izi:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Mapeto
Munkhaniyi, takambirana mavuto angapo okhudzana ndi ma vector mu fizikisi, kuyambira kuwonjezera ndi kuchotsa ma vector, scalar ndi cross multiplication, mpaka kukula kwa ma vector ndi resolution. Kumvetsetsa lingaliro ndi momwe ma vector amagwirira ntchito ndikofunikira mu fizikisi chifukwa zochitika zambiri zachilengedwe zitha kufotokozedwa pogwiritsa ntchito ma vector. Tikukhulupirira kuti mavuto awa a zitsanzo akuthandizani kumvetsetsa bwino lingaliro la ma vector.
