Imibuzo Eyisibonelo Ekhuluma Ngezivektha Nemisebenzi Yazo
Ama-vector angumqondo oyisisekelo kwizibalo kanye ne-physics, avame ukusetshenziswa emikhakheni eyahlukene yesayensi. Ama-vector amelela ubuningi ngobukhulu kanye nesiqondiso. Ngezansi kunezibonelo zezinkinga ezihilela ama-vector kanye nezingxoxo zemisebenzi eyahlukahlukene efana nokuhlanganisa, ukususa, kanye nokuphindaphinda ngama-scalar. Lesi sihloko sizokunikeza ukuqonda okujulile kokuthi ungaxazulula kanjani izinkinga ezihilela ama-vector.
1. Ukwengezwa kweVektha
Isibonelo Umbuzo 1
Kunikezwe amavekhtha amabili ngesimo sengxenye:
A = (3, 4)
B = (1, 2)
Bala umphumela wokwengeza amavekhtha u-A no-B.
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Ukwengezwa kwevektha kwenziwa ngokungeza izingxenye ezihambisanayo zamavektha amabili. Ngakho-ke, singabala
\[
A + B = (3 + 1, 4 + 2) = (4, 6)
\]
Ngakho-ke, umphumela wokwengeza amavekhtha u-A no-B u-(4, 6).
2. Ukususa amavektha
Isibonelo Umbuzo 2
Kunikezwe amavekhtha amabili ngesimo sengxenye:
C = (5, 7)
D = (2, 3)
Bala umphumela wokususa i-vector C ku-vector D.
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Ukususa amavektha kwenziwa ngokukhipha izingxenye ezihambisanayo zamavektha amabili. Ngakho-ke, singabala
\[
C – D = (5 – 2, 7 – 3) = (3, 4)
\]
Ngakho-ke, umphumela wokukhipha amavekhtha u-C no-D ungu-(3, 4).
3. Ukuphindaphinda kwamaVector ngamaScalars
Isibonelo Umbuzo 3
Ivektha enikeziwe u-E = (4, -2) kanye ne-scalar k = 3. Bala umphumela wokuphindaphinda ivektha u-E ngo-scalar k.
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Ukuphindaphinda kwevektha nge-scalar kwenziwa ngokuphindaphinda ingxenye ngayinye yevektha nge-scalar. Ngakho-ke, singabala
\[
k E = 3 (4, -2) = (3 4, 3 -2) = (12, -6)
\]
Ngakho-ke, umphumela wokuphindaphinda i-vector E nge-scalar k ngu-(12, -6).
4. Umkhiqizo we-Dot
Isibonelo Umbuzo 4
Kunikezwe amavekhtha amabili ngesimo sengxenye:
F = (1, 3)
G = (4, 2)
Bala umkhiqizo wamachashazi wamavektha u-F no-G.
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Umkhiqizo wamachashazi wamavektha amabili yisamba semikhiqizo yezingxenye zawo ezihambisanayo. Ngakho-ke, singabala
\[
F \cdot G = (1 4) + (3 2) = 4 + 6 = 10
\]
Ngakho-ke, umkhiqizo wamachashazi wamavektha u-F no-G ungu-10.
5. Umkhiqizo Ohlanganisiwe
Isibonelo Umbuzo 5
Njengoba kunikezwe amavekhtha amabili ku-3D:
H = (2, -3, 1)
I = (1, 4, -2)
Bala umkhiqizo ohlanganisiwe wamavektha u-H no-I.
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Umkhiqizo ohlanganisiwe wamavekhtha amabili anezinhlangothi ezintathu ukhiqizwa yi-determinant ye-matrix equkethe izingxenye zamavekhtha womabili. Ivekhtha ephumelayo inezingxenye ezilandelayo:
\[
H \izikhathi I = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 kanye no-3 kanye no-1 \\
1 kanye no-4 kanye no-2 \\
\end{vmatrix}
\]
Ngokubala isichazi, sithola:
\[
H \izikhathi I = (\mathbf{i}((-3)(-2) – (1)(4)) – \mathbf{j}(2(-2) – (1)(1)) + \mathbf{k}(2(4) – (-3)(1)))
\]
\[
= (\mathbf{i}(6 – 4) – \mathbf{j}(-4 – 1) + \mathbf{k}(8 + 3))
\]
\[
= (\mathbf{i}(2) – \mathbf{j}(-5) + \mathbf{k}(11))
\]
\[
= (2, 5, 11)
\]
Ngakho-ke, umkhiqizo ohlanganisiwe wamavekhtha u-H no-I ungu-(2, 5, 11).
6. Ubude noma Ubukhulu beVektha
Isibonelo Umbuzo 6
Uma unikezwe i-vector J = (6, 8). Bala ubude (ubukhulu) be-vector J.
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Ubude (ubukhulu) bevektha bubalwa kusetshenziswa ifomula:
\[
\| J \| = \sqrt{x^2 + y^2}
\]
Kulokhu, \( x = 6 \) kanye \( y = 8 \), ukuze:
\[
\| J \| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
Ngakho-ke, ubude (ubukhulu) bevektha u-J buyi-10.
7. I-Unit Vector
Isibonelo Umbuzo 7
Uma unikezwe ivektha K = (-5, 12). Thola ivektha yeyunithi ka-K.
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Ivektha yeyunithi iyivektha enobude obungu-1. Ukuze sithole ivektha yeyunithi yevektha, kumelwe sihlukanise ingxenye ngayinye yevektha ngobude (ubukhulu) bevektha. Ubude bevektha u-K bungabalwa kanje:
\[
\| K \|= \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Bese i-vector yeyunithi u-K ithi:
\[
\hat{K} = \left(\frac{-5}{13}, \frac{12}{13}\right)
\]
Ngakho-ke, ivektha yeyunithi yevektha K ingu-\(\left(\frac{-5}{13}, \frac{12}{13}\right)\).
Isiphetho
Ngezibonelo ezingenhla, sibone indlela amavekhtha kanye nemisebenzi yawo esebenza ngayo ezimweni ezahlukene. Ukwengeza nokususa amavekhtha kuhilela ukwengeza nokukhipha izingxenye ezihambisanayo. Ukuphindaphinda kwamavekhtha kungenziwa ngesimo somkhiqizo we-scalar noma wamachashazi, kanye nesimo somkhiqizo ophambene wamavekhtha e-3D. Singakwazi ngisho nokunquma ubude bevekhtha bese sithola i-vekhtha yayo yeyunithi.
Ukuqonda le mibono eyisisekelo kubalulekile ngoba ama-vector asetshenziswa ezinhlelweni eziningi emikhakheni ehlukahlukene, kufaka phakathi i-physics, ubunjiniyela, kanye nehluzo zekhompyutha. Ngokuzijwayeza okwanele, singaziqonda kahle lezi zenzo futhi sizisebenzise ekuhlaziyeni okuyinkimbinkimbi kakhulu nasekuxazululeni izinkinga.
