Imibuzo Yezibonelo kanye Nengxoxo Yamavektha Anezinhlangothi Ezintathu Kuhlelo Lokuxhumanisa lweCartesian
Amavekhtha anezinhlangothi ezintathu angumqondo obalulekile kwizibalo kanye nefiziksi, avame ukusetshenziselwa ukumela izinto noma izenzakalo esikhaleni esinezinhlangothi ezintathu. Kuhlelo lwe-Cartesian coordinate, lawa mavekhtha amelelwa yizingxenye ezintathu, ngokuvamile ezibizwa ngokuthi \( (x, y, z) \). Lesi sihloko sizoxoxa ngezibonelo eziningana zezinkinga kanye nezixazululo ezihlobene namavekhtha anezinhlangothi ezintathu ohlelweni lwe-Cartesian coordinate.
Ukuqonda Amavektha Anezinhlangothi Ezintathu
Ivektha esikhaleni esinezinhlangothi ezintathu ingachazwa ngokuthi \(\mathbf{A} = (A_x, A_y, A_z)\), lapho:
– \(A_x\) yingxenye yevektha eceleni kwe-x-axis.
– \(A_y\) yingxenye yevektha eceleni kwe-y-axis.
– \(A_z\) yingxenye yevektha eceleni kwe-z-axis.
Imibuzo Eyisibonelo Nengxoxo
Umbuzo 1: Umsebenzi Wokwengeza Amavektha
Uma unikezwe amavekhtha amabili, \(\mathbf{A} = (2, -3, 4)\) kanye \(\mathbf{B} = (-1, 5, 2)\). Bala isamba sala mavekhtha amabili.
Ingxoxo:
Ukwengezwa kwamavekhtha amabili \(\mathbf{A}\) kanye \(\mathbf{B}\) kwenziwa ngokungeza izingxenye zawo ezihambisanayo. Ngakho-ke, sinalokhu:
\[
\mathbf{C} = \mathbf{A} + \mathbf{B} = (A_x + B_x, A_y + B_y, A_z + B_z)
\]
Faka esikhundleni amanani evektha anikeziwe:
\[
\mathbf{C} = (2 + (-1), -3 + 5, 4 + 2) = (1, 2, 6)
\]
Ngakho-ke, umphumela wokwengeza amavekhtha \(\mathbf{A}\) kanye \(\mathbf{B}\) ngu \(\mathbf{C} = (1, 2, 6)\).
Umbuzo 2: Umsebenzi Wokususa Amavektha
Uma unikezwe amavekhtha amabili, \(\mathbf{A} = (4, 1, -2)\) kanye \(\mathbf{B} = (5, -3, 6)\). Bala ukususwa kwala mavekhtha amabili, okungukuthi \(\mathbf{A} – \mathbf{B}\).
Ingxoxo:
Ukususa amavekhtha amabili \(\mathbf{A}\) kanye \(\mathbf{B}\) kwenziwa ngokukhipha izingxenye zawo ezihambisanayo. Ngakho-ke, sinalokhu:
\[
\mathbf{D} = \mathbf{A} – \mathbf{B} = (A_x – B_x, A_y – B_y, A_z – B_z)
\]
Faka esikhundleni amanani evektha anikeziwe:
\[
\mathbf{D} = (4 – 5, 1 – (-3), -2 – 6) = (-1, 4, -8)
\]
Ngakho-ke, umphumela wokukhipha amavektha \(\mathbf{A}\) kanye \(\mathbf{B}\) ngu \(\mathbf{D} = (-1, 4, -8)\).
Umbuzo 3: Umsebenzi Wokuphindaphinda we-Scalar
Uma unikezwe i-vector \(\mathbf{A} = (3, -2, 7)\) kanye ne-scalar \(k = 4\). Bala umkhiqizo we-scalar walezi vector.
Ingxoxo:
Ukuphindaphinda i-scalar \(k\) nge-vector \(\mathbf{A}\) kwenziwa ngokuphindaphinda ingxenye ngayinye ye-vector ngaleyo scalar. Ngakho-ke, sinalokhu:
\[
\mathbf{E} = k \cdot \mathbf{A} = k \cdot (A_x, A_y, A_z) = (k \cdot A_x, k \cdot A_y, k \cdot A_z)
\]
Faka amanani anikeziwe esikhundleni sawo:
\[
\mathbf{E} = 4 \cdot (3, -2, 7) = (4 \cdot 3, 4 \cdot -2, 4 \cdot 7) = (12, -8, 28)
\]
Ngakho-ke, umphumela wokuphindaphinda i-scalar \(k\) nge-vector \(\mathbf{A}\) ngu-\(\mathbf{E} = (12, -8, 28)\).
Umbuzo 4: Ubude beVektha
Bala ubude (ubukhulu) bevektha \(\mathbf{A} = (1, 2, 2)\).
Ingxoxo:
Ubude noma ubukhulu bevektha \(\mathbf{A} = (A_x, A_y, A_z)\) bungabalwa kusetshenziswa ifomula:
\[
|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
\]
Faka amanani anikeziwe esikhundleni sawo:
\[
|\mathbf{A}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
Ngakho-ke, ubude bevektha \(\mathbf{A}\) bungu-3.
Umbuzo 5: Umkhiqizo we-Dot
Uma unikezwe amavekhtha amabili, \(\mathbf{A} = (1, 0, -1)\) kanye \(\mathbf{B} = (2, 3, 4)\). Bala umkhiqizo wamachashazi wala mavekhtha amabili.
Ingxoxo:
Umkhiqizo wamachashazi wamavekhtha amabili \(\mathbf{A} = (A_x, A_y, A_z)\) kanye \(\mathbf{B} = (B_x, B_y, B_z)\) wenziwa ngokuphindaphinda izingxenye ezihambisanayo bese uzingeza. Ngakho-ke, sinalokhu:
\[
\mathbf{A} \cdot \mathbf{B} = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z
\]
Faka amanani anikeziwe esikhundleni sawo:
\[
\mathbf{A} \cdot \mathbf{B} = (1 \cdot 2) + (0 \cdot 3) + (-1 \cdot 4) = 2 + 0 – 4 = -2
\]
Ngakho-ke, umkhiqizo wamachashazi wamavektha \(\mathbf{A}\) kanye \(\mathbf{B}\) ungu--2.
Umbuzo 6: Umkhiqizo Ohlanganisiwe
Uma unikezwe amavekhtha amabili, \(\mathbf{A} = (1, 2, 3)\) kanye \(\mathbf{B} = (4, 5, 6)\). Bala umkhiqizo ohlanganisiwe walezi vekhtha ezimbili.
Ingxoxo:
Umkhiqizo ohlanganisiwe wamavekhtha amabili \(\mathbf{A} = (A_x, A_y, A_z)\) kanye \(\mathbf{B} = (B_x, B_y, B_z)\) wenziwa kusetshenziswa ifomula elandelayo:
\[
\mathbf{A} \times \mathbf{B} = \left( (A_y \cdot B_z – A_z \cdot B_y), (A_z \cdot B_x – A_x \cdot B_z), (A_x \cdot B_y – A_y \cdot B_x) \right)
\]
Faka amanani anikeziwe esikhundleni sawo:
\[
\mathbf{A} \times \mathbf{B} = \left( (2 \cdot 6 – 3 \cdot 5), (3 \cdot 4 – 1 \cdot 6), (1 \cdot 5 – 2 \cdot 4) \right) = (12 – 15, 12 – 6, 5 – 8) = (-3, 6, -3)
\]
Ngakho-ke, umkhiqizo ohlanganisiwe wamavektha \(\mathbf{A}\) kanye \(\mathbf{B}\) ngu \(\mathbf{A} \times \mathbf{B} = (-3, 6, -3)\).
Isiphetho
Amavekhtha anezinhlangothi ezintathu ohlelweni lwe-Cartesian coordinate angamathuluzi abalulekile emikhakheni ehlukahlukene yesayensi nobunjiniyela. Ngezibonelo nezingxoxo ezingenhla, sibone ukuthi singawenza kanjani imisebenzi ehlukahlukene eyisisekelo kumavekhtha, njengokuhlanganisa, ukususa, ukuphindaphinda kwe-scalar, kanye nemikhiqizo yamachashazi kanye ne-cross. Ukuqonda okuqinile kwale mibono kuzoba usizo kakhulu hhayi kuphela kwizibalo kodwa futhi nasekusetshenzisweni okusebenzayo ku-physics, ubunjiniyela, kanye nesayensi yekhompyutha.
