Imibuzo Eyisibonelo Exoxa Ngamavektha Amabili Kuhlelo Lokuhlanganisa
I-vector iyinani elinobukhulu kanye nesiqondiso. Ama-vector avame ukusetshenziswa ezihlokweni ezahlukene zezibalo kanye ne-physics ukumela izimo ezahlukahlukene. Kulesi sihloko, sizoxoxa ngezibonelo zama-vector anezinhlangothi ezimbili ohlelweni oluhambisanayo.
Imiqondo Eyisisekelo Yama-Vector Ezinhlelweni Zokuxhumanisa
Ivektha ohlelweni lokuhlanganisa olunezinhlangothi ezimbili ingamelwa njenge-\(\vec{A} = (a_1, a_2)\), lapho i-\(a_1\) iyi-x-component yevektha kanye ne-\(a_2\) iyi-y-component yevektha. Ivektha ingahlukaniswa ibe yizingxenye ezimbili, okuyi-x-component kanye ne-y-component.
Ukwengezwa Nokususwa Kwevekhtha
Ukwengezwa kwamavektha amabili \(\vec{A} = (a_1, a_2)\) kanye \(\vec{B} = (b_1, b_2)\) yilokhu:
\[
\vec{A} + \vec{B} = (a_1 + b_1, a_2 + b_2)
\]
Ngenkathi ukuncishiswa kungukuthi:
\[
\vec{A} – \vec{B} = (a_1 – b_1, a_2 – b_2)
\]
Ukuphindaphinda kwe-Scalar
Uma i-\(\vec{A} = (a_1, a_2)\) kanye ne-\(k\) iyi-scalar, khona-ke i-\(k\vec{A}\) iyi:
\[
k\vec{A} = (k \cdot a_1, k \cdot a_2)
\]
Ubukhulu beVektha
Ubukhulu noma ubude bevektha \(\vec{A} = (a_1, a_2)\) ngu:
\[
|\vec{A}| = \sqrt{a_1^2 + a_2^2}
\]
I-Unit Vector
Ivektha yeyunithi iyivektha enobude beyunithi eyodwa. Ivektha yeyunithi ye-\(\vec{A} = (a_1, a_2)\) ithi:
\[
\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \left( \frac{a_1}{\sqrt{a_1^2 + a_2^2}}, \frac{a_2}{\sqrt{a_1^2 + a_2^2}} \right)
\]
Imibuzo Eyisibonelo Nengxoxo
Umbuzo 1: Ukwengeza nokususa amaVektha
Amavektha amabili anikezwe kanje: \(\vec{A} = (3, 4)\) kanye \(\vec{B} = (1, 2)\). Thola umkhiqizo we-\(\vec{A} + \vec{B}\) kanye ne-\(\vec{A} – \vec{B}\).
Ingxoxo:
\[
\vec{A} + \vec{B} = (3 + 1, 4 + 2) = (4, 6)
\]
\[
\vec{A} – \vec{B} = (3 – 1, 4 – 2) = (2, 2)
\]
Umbuzo 2: Ukuphindaphinda kwe-Scalar
Uma unikezwe i-vector \(\vec{C} = (2, -3)\), bala \(3\vec{C}\) kanye \(-2\vec{C}\).
Ingxoxo:
\[
3\vec{C} = 3 \cdot (2, -3) = (6, -9)
\]
\[
-2\vec{C} = -2 \cdot (2, -3) = (-4, 6)
\]
Umbuzo 3: Ubukhulu beVektha
Bala ubukhulu bevektha \(\vec{D} = (5, 12)\).
Ingxoxo:
\[
|\vec{D}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Umbuzo 4: Ama-Unit Vectors
Thola i-vector yeyunithi yevektha \(\vec{E} = (4, 3)\).
Ingxoxo:
\[
|\vec{E}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
\[
\hat{E} = \frac{\vec{E}}{|\vec{E}|} = \left( \frac{4}{5}, \frac{3}{5} \right)
\]
Umbuzo 5: Indawo kanye nebanga lamaVektha
Amaphuzu amabili endizeni yokuxhumanisa enezinhlangothi ezimbili yi-P(2, 3) kanye ne-Q(5, 7). Nquma i-vector yesikhundla kusukela ephuzwini P kuya ephuzwini Q kanye nebanga eliphakathi kwawo.
Ingxoxo:
Ivektha yesikhundla kusukela ku-P kuya ku-Q ithi:
\[
\vec{PQ} = \vec{Q} – \vec{P} = (5 – 2, 7 – 3) = (3, 4)
\]
Ibanga eliphakathi kwamaphuzu u-P no-Q lithi:
\[
|\vec{PQ}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Umbuzo 6: Umphumela Womkhiqizo We-Dot
Uma \(\vec{F} = (-3, 4)\) kanye \(\vec{G} = (2, 1)\), bala umkhiqizo wechashazi we \(\vec{F} \cdot \vec{G}\).
Ingxoxo:
Umkhiqizo wamachashazi wamavektha amabili uthi:
\[
\vec{F} \cdot \vec{G} = (-3) \cdot 2 + 4 \cdot 1 = -6 + 4 = -2
\]
Umbuzo 7: I-engela Phakathi Kwamavektha Amabili
Uma \(\vec{H} = (7, -4)\) kanye \(\vec{I} = (3, 0)\), nquma i-engeli phakathi kwamavektha amabili.
Ingxoxo:
Ukuze sinqume i-engeli phakathi kwama-vector amabili, sisebenzisa ifomula:
\[
\cos \theta = \frac{\vec{H} \cdot \vec{I}}{|\vec{H}| |\vec{I}|}
\]
Okokuqala, bala umkhiqizo wechashazi \(\vec{H} \cdot \vec{I}\):
\[
\vec{H} \cdot \vec{I} = 7 \cdot 3 + (-4) \cdot 0 = 21 + 0 = 21
\]
Bese, bala ubukhulu be-\(\vec{H}\) kanye ne-\(\vec{I}\):
\[
|\vec{H}| = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}
\]
\[
|\vec{I}| = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]
Faka la manani kufomula:
\[
\cos \theta = \frac{21}{\sqrt{65} \cdot 3} = \frac{21}{3\sqrt{65}} = \frac{7}{\sqrt{65}}
\]
Ngakho-ke, \(\theta = \cos^{-1} \left( \frac{7}{\sqrt{65}} \right) \).
Umbuzo 8: Ukubikezela kweVektha
Kumavektha \(\vec{J} = (2, 1)\) kanye \(\vec{K} = (-1, 3)\), bala ukuvela kwe-\(\vec{J}\) ku-\(\vec{K}\).
Ingxoxo:
Ukuqagela kwe-\(\vec{J}\) ku-\(\vec{K}\) kungukuthi:
\[
\umbhalo{proj}_{\vec{K}} \vec{J} = \left( \frac{\vec{J} \cdot \vec{K}}{|\vec{K}|^2} \kwesokudla) \vec{K}
\]
Okokuqala, bala umkhiqizo wechashazi \(\vec{J} \cdot \vec{K}\):
\[
\vec{J} \cdot \vec{K} = 2 \cdot (-1) + 1 \cdot 3 = -2 + 3 = 1
\]
Bese, ubukhulu be-\(\vec{K}\):
\[
|\vec{K}| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]
Ukuze,:
\[
|\vec{K}|^2 = 10
\]
Faka ifomula:
\[
\text{proj}_{\vec{K}} \vec{J} = \left( \frac{1}{10} \right) \vec{K} = \left( \frac{1}{10} \right) (-1, 3) = \left( -\frac{1}{10}, \frac{3}{10} \right)
\]
Lezi yizibonelo zezinkinga nezingxoxo ezihlobene nama-vector anezinhlangothi ezimbili ohlelweni oluhlanganisayo. Ukuqonda kahle ama-vector kungaba usizo ekusetshenzisweni okuningi kwezibalo, i-physics, kanye nobunjiniyela. Ukuzijwayeza ngezibonelo ezahlukahlukene kungajulisa ukuqonda kwakho ngalo mqondo, kuvumele ukuthi usetshenziswe ngempumelelo ezimweni ezahlukahlukene.
