Zitsanzo za Mafunso Okhudza Ma Vector ndi Ntchito Zawo
Maveketa ndi mfundo yofunikira kwambiri mu masamu ndi fizikisi, yomwe imagwiritsidwa ntchito nthawi zambiri m'magawo osiyanasiyana asayansi. Maveketa amaimira kuchuluka kwa zinthu zomwe zili ndi kukula ndi njira. Pansipa pali zitsanzo za mavuto okhudzana ndi maveketa ndi zokambirana za ntchito zosiyanasiyana monga kuwonjezera, kuchotsa, ndi kuchulukitsa ndi ma scalar. Nkhaniyi ipereka kumvetsetsa kwakuya kwa momwe mungathetsere mavuto okhudzana ndi maveketa.
1. Kuwonjezera Vekitala
Chitsanzo cha Funso 1
Popeza ma vector awiri ali mu mawonekedwe a gawo:
A = (3, 4)
B = (1, 2)
Werengerani zotsatira za kuwonjezera ma vector A ndi B.
Zokambirana
Kuwonjezera mavekitala kumachitika powonjezera zigawo zofanana za mavekitala awiriwa. Motero, tikhoza kuwerengera
\[
A + B = (3 + 1, 4 + 2) = (4, 6)
\]
Kotero, zotsatira za kuwonjezera ma vector A ndi B ndi (4, 6).
2. Kuchotsa Vekitala
Chitsanzo cha Funso 2
Popeza ma vector awiri ali mu mawonekedwe a gawo:
C = (5, 7)
D = (2, 3)
Werengerani zotsatira za kuchotsa vekitala C kuchokera ku vekitala D.
Zokambirana
Kuchotsa mavekitala kumachitika pochotsa zigawo zofanana za mavekitala awiriwa. Motero, tikhoza kuwerengera
\[
C – D = (5 – 2, 7 – 3) = (3, 4)
\]
Kotero, zotsatira za kuchotsa ma vector C ndi D ndi (3, 4).
3. Kuchulukitsa kwa Ma Vector ndi Ma Scalars
Chitsanzo cha Funso 3
Vekitala yoperekedwa E = (4, -2) ndi scalar k = 3. Werengani zotsatira za kuchulukitsa vekitala E ndi scalar k.
Zokambirana
Kuchulukitsa kwa vekitala ndi scalar kumachitika pochulukitsa gawo lililonse la vekitala ndi scalar. Chifukwa chake, titha kuwerengera
\[
k E = 3 (4, -2) = (3 4, 3 -2) = (12, -6)
\]
Chifukwa chake, zotsatira za kuchulukitsa vekitala E ndi scalar k ndi (12, -6).
4. Katundu wa Dot
Chitsanzo cha Funso 4
Popeza ma vector awiri ali mu mawonekedwe a gawo:
F = (1, 3)
G = (4, 2)
Werengerani zotsatira za dot za ma vector F ndi G.
Zokambirana
Chopangidwa ndi dot cha ma vector awiri ndi chiwonkhetso cha zinthu zomwe zili m'zigawo zawo zofanana. Chifukwa chake, titha kuwerengera
\[
F \cdot G = (1 4) + (3 2) = 4 + 6 = 10
\]
Kotero, zotsatira za dot za ma vector F ndi G ndi 10.
5. Zogulitsa Zosiyanasiyana
Chitsanzo cha Funso 5
Popeza ma vector awiri mu 3D:
H = (2, -3, 1)
Ine = (1, 4, -2)
Werengerani zotsatira zosakanikirana za ma vector H ndi I.
Zokambirana
Chopangidwa chophatikizana cha mavekitala awiri amitundu itatu chimapangidwa ndi chodziwikiratu cha matrix chomwe chili ndi zigawo za mavekitala onse awiri. Vekitala yotsatira ili ndi zigawo izi:
\[
H \times I = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 ndi -3 ndi 1 \\
1 & 4 & -2 \\
\end{vmatrix}
\]
Mwa kuwerengera chodziwikiratu, timapeza:
\[
H \times I = (\mathbf{i}((-3)(-2) – (1)(4)) – \mathbf{j}(2(-2) – (1)(1)) + \mathbf{k}(2(4) – (-3)(1)))
\]
\[
= (\mathbf{i}(6 – 4) – \mathbf{j}(-4 – 1) + \mathbf{k}(8 + 3))
\]
\[
= (\mathbf{i}(2) – \mathbf{j}(-5) + \mathbf{k}(11))
\]
\[
= (2, 5, 11)
\]
Kotero, zotsatira zosakanikirana za ma vector H ndi I ndi (2, 5, 11).
6. Kutalika kapena Kukula kwa Vekitala
Chitsanzo cha Funso 6
Kupatsidwa vekitala J = (6, 8). Werengani kutalika (kukula) kwa vekitala J.
Zokambirana
Kutalika (kukula) kwa vekitala kumawerengedwa pogwiritsa ntchito fomula iyi:
\[
\| J \| = \sqrt{x^2 + y^2}
\]
Pankhaniyi, \( x = 6 \) ndi \( y = 8 \), kotero kuti:
\[
\| J \| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
Kotero, kutalika (kukula) kwa vekitala J ndi 10.
7. Vekitala ya Unit
Chitsanzo cha Funso 7
Popeza vekitala K = (-5, 12). Pezani vekitala ya unit ya K.
Zokambirana
Vekitala ya unit ndi vekitala yomwe ili ndi kutalika kwa 1. Kuti tipeze vekitala ya unit ya vekitala, tiyenera kugawa gawo lililonse la vekitala ndi kutalika (kukula) kwa vekitala. Kutalika kwa vekitala K kumatha kuwerengedwa motere:
\[
\| K \|= \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Kenako vector ya unit K ndi:
\[
\hat{K} = \left(\frac{-5}{13}, \frac{12}{13}\right)
\]
Kotero, vekitala ya vekitala K ndi \(\left(\frac{-5}{13}, \frac{12}{13}\right)\).
Mapeto
Kudzera mu zitsanzo zomwe zili pamwambapa, taona momwe mavekitala ndi ntchito zawo zimagwirira ntchito m'malo osiyanasiyana. Kuwonjezera ndi kuchotsa mavekitala kumaphatikizapo kuwonjezera ndi kuchotsa zigawo zofanana. Kuchulukitsa mavekitala kungachitike mu mawonekedwe a chinthu cha scalar kapena dot, komanso mu mawonekedwe a chinthu chophatikizana cha mavekitala a 3D. Titha kudziwa kutalika kwa vekitala ndikupeza vekitala yake ya unit.
Kumvetsetsa mfundo zazikuluzikuluzi n'kofunika kwambiri chifukwa ma vector amagwiritsidwa ntchito m'magawo osiyanasiyana, kuphatikizapo fizikisi, uinjiniya, ndi zithunzi zamakompyuta. Ndi machitidwe okwanira, titha kudziwa bwino ntchito izi ndikuzigwiritsa ntchito pakusanthula kovuta komanso kuthetsa mavuto.
