Zitsanzo za Mafunso Okambirana za Ma Vector Awiri mu Coordinate System
Vektha ndi kuchuluka komwe kuli ndi kukula ndi chitsogozo. Vektha amagwiritsidwa ntchito nthawi zambiri m'mitu yosiyanasiyana ya masamu ndi fizikisi kuti ayimire zochitika zosiyanasiyana. M'nkhaniyi, tikambirana zitsanzo za vektha ziwiri mu dongosolo logwirizana.
Malingaliro Oyambira a Ma Vector mu Coordinate Systems
Vekitala mu dongosolo la ma coordinate awiriawiri ikhoza kuimiridwa ngati \(\vec{A} = (a_1, a_2)\), pomwe \(a_1\) ndiye x-component ya vekitala ndipo \(a_2\) ndiye y-component ya vekitala. Vekitala ikhoza kugawidwa m'magawo awiri, omwe ndi x-component ndi y-component.
Kuwonjezera ndi Kuchotsa Vekitala
Kuwonjezera ma vector awiri \(\vec{A} = (a_1, a_2)\) ndi \(\vec{B} = (b_1, b_2)\) ndi:
\[
\vec{A} + \vec{B} = (a_1 + b_1, a_2 + b_2)
\]
Ngakhale kuchepetsa kuli:
\[
\vec{A} – \vec{B} = (a_1 – b_1, a_2 – b_2)
\]
Kuchulukitsa kwa Scalar
Ngati \(\vec{A} = (a_1, a_2)\) ndi \(k\) ndi scalar, ndiye kuti \(k\vec{A}\) ndi:
\[
k\vec{A} = (k \cdot a_1, k \cdot a_2)
\]
Kukula kwa Vekitala
Kukula kapena kutalika kwa vekitala \(\vec{A} = (a_1, a_2)\) ndi:
\[
|\vec{A}| = \sqrt{a_1^2 + a_2^2}
\]
Vekitala ya Unit
Vekitala ya unit ndi vekitala yomwe ili ndi utali wa unit imodzi. Vekitala ya unit ya \(\vec{A} = (a_1, a_2)\) ndi:
\[
\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \left( \frac{a_1}{\sqrt{a_1^2 + a_2^2}}, \frac{a_2}{\sqrt{a_1^2 + a_2^2}} \right)
\]
Mafunso ndi Kukambirana Zitsanzo
Funso 1: Kuwonjezera ndi Kuchotsa Ma Vector
Ma vector awiri aperekedwa motere: \(\vec{A} = (3, 4)\) ndi \(\vec{B} = (1, 2)\). Pezani chinthu cha \(\vec{A} + \vec{B}\) ndi \(\vec{A} – \vec{B}\).
Kukambirana:
\[
\vec{A} + \vec{B} = (3 + 1, 4 + 2) = (4, 6)
\]
\[
\vec{A} – \vec{B} = (3 – 1, 4 – 2) = (2, 2)
\]
Funso 2: Kuchulukitsa kwa Scalar
Popeza vekitala \(\vec{C} = (2, -3)\), werengerani \(3\vec{C}\) ndi \(-2\vec{C}\).
Kukambirana:
\[
3\vec{C} = 3 \cdot (2, -3) = (6, -9)
\]
\[
-2\vec{C} = -2 \cdot (2, -3) = (-4, 6)
\]
Funso 3: Kukula kwa Vekitala
Werengerani kukula kwa vekitala \(\vec{D} = (5, 12)\).
Kukambirana:
\[
|\vec{D}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Funso 4: Ma Vekitala a Zigawo
Pezani vekitala ya vekitala \(\vec{E} = (4, 3)\).
Kukambirana:
\[
|\vec{E}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
\[
\hat{E} = \frac{\vec{E}}{|\vec{E}|} = \left( \frac{4}{5}, \frac{3}{5} \right)
\]
Funso 5: Malo ndi Mtunda wa Ma Vector
Mfundo ziwiri mu ndege ya coordinate ya magawo awiri ndi P(2, 3) ndi Q(5, 7). Dziwani vekitala ya malo kuchokera pa mfundo P kupita pa mfundo Q ndi mtunda pakati pawo.
Kukambirana:
Vekitala ya malo kuyambira P mpaka Q ndi:
\[
\vec{PQ} = \vec{Q} – \vec{P} = (5 – 2, 7 – 3) = (3, 4)
\]
Mtunda pakati pa mfundo P ndi Q ndi:
\[
|\vec{PQ}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Funso 6: Zotsatira za Dot Product
Ngati \(\vec{F} = (-3, 4)\) ndi \(\vec{G} = (2, 1)\), werengerani zotsatira za dot ya \(\vec{F} \cdot \vec{G}\).
Kukambirana:
Chotsatira cha madontho a ma vector awiri ndi:
\[
\vec{F} \cdot \vec{G} = (-3) \cdot 2 + 4 \cdot 1 = -6 + 4 = -2
\]
Funso 7: Mng'alu Pakati pa Ma Vector Awiri
Ngati \(\vec{H} = (7, -4)\) ndi \(\vec{I} = (3, 0)\), dziwani ngodya pakati pa mavekitala awiriwa.
Kukambirana:
Kuti tidziwe ngodya pakati pa ma vector awiri, timagwiritsa ntchito fomula iyi:
\[
\cos \theta = \frac{\vec{H} \cdot \vec{I}}{|\vec{H}| |\vec{I}|}
\]
Choyamba, werengerani chinthu cha dot \(\vec{H} \cdot \vec{I}\):
\[
\vec{H} \cdot \vec{I} = 7 \cdot 3 + (-4) \cdot 0 = 21 + 0 = 21
\]
Kenako, werengani kukula kwa \(\vec{H}\) ndi \(\vec{I}\):
\[
|\vec{H}| = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}
\]
\[
|\vec{I}| = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]
Lowetsani mfundo izi mu fomula:
\[
\cos \theta = \frac{21}{\sqrt{65} \cdot 3} = \frac{21}{3\sqrt{65}} = \frac{7}{\sqrt{65}}
\]
Kotero, \(\theta = \cos^{-1} \left( \frac{7}{\sqrt{65}} \right) \).
Funso 8: Kuwonetsera kwa Vekitala
Pa ma vectors \(\vec{J} = (2, 1)\) ndi \(\vec{K} = (-1, 3)\), werengerani pulojekiti ya \(\vec{J}\) pa \(\vec{K}\).
Kukambirana:
Kuwonetsera kwa \(\vec{J}\) pa \(\vec{K}\) ndi:
\[
\text{proj}_{\vec{K}} \vec{J} = \left( \frac{\vec{J} \cdot \vec{K}}{|\vec{K}|^2} \right) \vec{K}
\]
Choyamba, werengerani chinthu cha dot \(\vec{J} \cdot \vec{K}\):
\[
\vec{J} \cdot \vec{K} = 2 \cdot (-1) + 1 \cdot 3 = -2 + 3 = 1
\]
Kenako, kukula kwa \(\vec{K}\):
\[
|\vec{K}| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]
Ndicholinga choti,:
\[
|\vec{K}|^2 = 10
\]
Lowani mu fomula:
\[
\text{proj}_{\vec{K}} \vec{J} = \left( \frac{1}{10} \right) \vec{K} = \left( \frac{1}{10} \right) (-1, 3) = \left( -\frac{1}{10}, \frac{3}{10} \right)
\]
Izi ndi zitsanzo za mavuto ndi zokambirana zokhudzana ndi ma vector a mbali ziwiri mu dongosolo logwirizana. Kumvetsetsa bwino ma vector kungathandize pa ntchito zambiri mu masamu, fizikisi, ndi uinjiniya. Kuchita masewera olimbitsa thupi ndi zitsanzo zosiyanasiyana kungakuthandizeni kumvetsetsa bwino lingaliro ili, zomwe zingakupatseni mwayi woti ligwiritsidwe ntchito bwino pazochitika zosiyanasiyana.
