Isibonelo Semibuzo Yengxoxo Yokusebenza Kwevektha
Imisebenzi ye-Vector ingumqondo oyisisekelo kwizibalo ovame ukuvela emikhakheni eyahlukene yocwaningo, njenge-physics, ubunjiniyela, kanye nesayensi yekhompyutha. Kulesi sihloko, sizoxoxa ngezibonelo eziningana zokusebenza kwe-vector kanye nezixazululo zazo ukuze sinikeze ukuqonda okujulile nokuqondile. Lezi zibonelo zizohlanganisa imisebenzi eyisisekelo njengokwengeza nokususa kwe-vector, kanye nemisebenzi ethuthukile kakhulu njengokuphindaphinda kwe-scalar kanye nokuphindaphinda kwe-cross-vector.
1. Ukwengezwa Nokususwa Kwevektha
Isibonelo Umbuzo 1
Kunikezwe amavekhtha amabili u-A no-B ngesimo sengxenye:
\[ \mathbf{A} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \]
\[ \mathbf{B} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} \]
Bala umphumela wokwengeza nokukhipha amavekhtha amabili.
Ingxoxo
Ukuze singeze amavektha, sengeza ingxenye ngayinye ehambisanayo yamavektha amabili.
\[ \mathbf{A} + \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \\ -1 + 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 1 \end{pmatrix} \]
Ukuze sikhiphe amavektha, sisusa ingxenye ngayinye ehambisanayo yamavektha womabili.
\[ \mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} – \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 – (-1) \\ 3 – 4 \\ -1 – 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ -3 \end{pmatrix} \]
2. Ukuphindaphinda kwe-Scalar nge-Vector
Isibonelo Umbuzo 2
Uma unikezwe i-vector C kanye ne-scalar k:
\[ \mathbf{C} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \]
\[k = 4 \]
Bala umkhiqizo we-scalar we-vector C nge-scalar k.
Ingxoxo
Ukuphindaphinda kwe-scalar yi-vector kwenziwa ngokuphindaphinda ingxenye ngayinye ye-vector yi-scalar.
\[ k \mathbf{C} = 4 \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \cdot 1 \\ 4 \cdot (-2) \\ 4 \cdot 3 \end{pmatrix} = \begin{pmatrix} 4 \\ -8 \\ 12 \end{pmatrix} \]
3. Umkhiqizo we-Dot
Isibonelo Umbuzo 3
Njengoba kunikezwe amavekhtha amabili u-D no-E:
\[ \mathbf{D} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} \]
\[ \mathbf{E} = \qala{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \]
Bala umkhiqizo wamachashazi wamavektha amabili.
Ingxoxo
Umkhiqizo wamachashazi wamavektha amabili utholakala ngokungeza imikhiqizo yezingxenye zawo ezihambisanayo.
\[ \mathbf{D} \cdot \mathbf{E} = 3 \cdot 1 + (-2) \cdot 0 + 4 \cdot (-1) = 3 + 0 – 4 = -1 \]
4. Umkhiqizo Ohlanganisiwe
Isibonelo Umbuzo 4
Njengoba kunikezwe amavekhtha amabili u-F no-G:
\[ \mathbf{F} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \]
\[ \mathbf{G} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \]
Bala umkhiqizo ohlanganisiwe wamavektha amabili.
Ingxoxo
Umkhiqizo ohlanganisiwe wamavekhtha amabili esikhaleni esinezinhlangothi ezintathu utholakala ngokusebenzisa isichazi se-matrix esakhiwe yilawo mavekhtha. Umkhiqizo ohlanganisiwe unikezwa yifomula:
\[ \mathbf{F} \izikhathi \mathbf{G} = \qala{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \end{vmatrix} \]
Lokhu kungabalwa ngale ndlela elandelayo:
\[
\mathbf{F} \izikhathi \mathbf{G} = \mathbf{i} \qala{vmatrix} 3 & 4 \\ -1 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} +k 2\matrix &1\matrix &1\2\matrix & -1 \end{vmatrix}
\]
Ukubala isichazamazwi se-submatrix ngayinye:
\[
= \mathbf{i} (3 \cdot 2 – 4 \cdot -1) – \mathbf{j} (2 \cdot 2 – 4 \cdot 1) + \mathbf{k} (2 \cdot -1 – 3 \cdot 1)
\]
\[
= \mathbf{i} (6 + 4) – \mathbf{j} (4 – 4) + \mathbf{k} (-2 – 3)
\]
\[
= \mathbf{i} (10) – \mathbf{j} (0) + \mathbf{k} (-5)
\]
\[
= \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}
\]
Ngakho-ke, umkhiqizo ohlanganisiwe we-F kanye ne-G uthi:
\[ \mathbf{F} \times \mathbf{G} = \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix} \]
5. Ukunquma i-Engela phakathi kwamaVektha Amabili
Isibonelo Umbuzo 5
Njengoba kunikezwe amavekhtha amabili u-H no-I:
\[ \mathbf{H} = \begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{I} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} \]
Thola i-engeli phakathi kwamavektha amabili.
Ingxoxo
I-engeli \(\theta\) phakathi kwamavekhtha amabili ingatholakala ngokusebenzisa ubudlelwano phakathi komkhiqizo wechashazi kanye nobukhulu bamavekhtha amabili:
\[ \mathbf{H} \cdot \mathbf{I} = \| \mathbf{H} \| \| \mathbf{I} \| \cos \theta \]
Okokuqala, bala umkhiqizo wechashazi \( \mathbf{H} \cdot \mathbf{I} \):
\[ \mathbf{H} \cdot \mathbf{I} = 6 \cdot 1 + 2 \cdot 4 + 3 \cdot (-2) = 6 + 8 – 6 = 8 \]
Okulandelayo, bala ubukhulu bamavektha womabili:
\[ \| \mathbf{H} \| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]
\[ \| \mathbf{I} \| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \]
Bese, faka la manani esikhundleni sefomula ye-angle:
\[ \cos \theta = \frac{\mathbf{H} \cdot \mathbf{I}}{\| \mathbf{H} \| \| \mathbf{I} \|} = \frac{8}{7\sqrt{21}} \]
\[ \theta = \cos^{-1} \kwesobunxele( \frac{8}{7\sqrt{21}} \kwesokudla) \]
Njengomphumela wokugcina, singasebenzisa i-calculator ukuthola inani le-engeli:
\[ \theta \cishe 73,4^\circ \]
Isiphetho
Umqondo wokusebenza kwe-vector ubalulekile kwizibalo nesayensi. Lesi sihloko sixoxa ngezinkinga eziningana zezibonelo kanye nezixazululo zazo, kusukela ekuhlanganiseni nasekususeni kwe-vector, ekuphindaphindeni kwe-scalar, kumkhiqizo wamachashazi, kumkhiqizo owela, kanye nasekunqumeni i-engeli phakathi kwama-vector amabili. Ngokusebenzisa lezi zibonelo, sithemba ukuthuthukisa ukuqonda kwakho imisebenzi ye-vector futhi sikusize uxazulule izinkinga ezihilela ama-vector ezimweni ezahlukene.
