Muenzaniso weMibvunzo yeKukurukurirana kweKushanda kweVector
Mashandiro eVector ipfungwa huru mumasvomhu inowanzoonekwa munzvimbo dzakasiyana dzekutsvaga, dzakadai sefizikisi, mainjiniya, uye sainzi yemakombiyuta. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yemabasa evector nemhinduro dzawo kuti tipe kunzwisisa kwakadzama uye kwakasimba. Mienzaniso iyi ichafukidza mashandiro ekutanga akadai sekuwedzera nekubvisa vector, pamwe nemabasa epamusoro akadai sekuwedzera kwescalar uye kuwanda kwevector.
1. Kuwedzera nekubvisa mavekita
Muenzaniso Mubvunzo 1
Zvichipiwa mavector maviri A naB muchimiro chechikamu:
\[ \mathbf{A} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \]
\[ \mathbf{B} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} \]
Verenga mhedzisiro yekuwedzera nekubvisa kwemavector maviri.
Kukurukurirana
Kuti tiwedzere vhector, tinowedzera chikamu chimwe nechimwe chinoenderana nemavector maviri aya.
\[ \mathbf{A} + \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \\ -1 + 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 1 \end{pmatrix} \]
Pakubvisa vhekita, tinobvisa chikamu chimwe nechimwe chinoenderana nemavhekita ese ari maviri.
\[ \mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} – \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 – (-1) \\ 3 – 4 \\ -1 – 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ -3 \end{pmatrix} \]
2. Kuwanda kweScalar neVector
Muenzaniso Mubvunzo 2
Kupiwa vhekitari C uye scalar k:
\[ \mathbf{C} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \]
\[k = 4 \]
Verenga chigadzirwa che scalar che vector C ne scalar k .
Kukurukurirana
Kuwanda kwe scalar ne vector kunoitwa nekuwanza chikamu chimwe nechimwe che vector ne 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. Chigadzirwa cheDot
Muenzaniso Mubvunzo 3
Zvichipiwa mavector maviri D na E:
\[ \mathbf{D} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} \]
\[ \mathbf{E} = \kutanga{pmatrix} 1 \\ 0 \\ -1 \kuguma{pmatrix} \]
Verenga mhedzisiro yemadonhwe emaveki maviri aya.
Kukurukurirana
Chigadzirwa chemadonhwe chevectors maviri chinowanikwa nekuwedzera zvigadzirwa zvezvikamu zvavo zvinoenderana.
\[ \mathbf{D} \cdot \mathbf{E} = 3 \cdot 1 + (-2) \cdot 0 + 4 \cdot (-1) = 3 + 0 – 4 = -1 \]
4. Chigadzirwa Chinosiyana
Muenzaniso Mubvunzo 4
Zvichipiwa mavector maviri F na G:
\[ \mathbf{F} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \]
\[ \mathbf{G} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \]
Verengai mhedzisiro ye mavector maviri aya.
Kukurukurirana
Chigadzirwa che mavector maviri ari munzvimbo ine mativi matatu chinowanikwa nekushandisa chinongedzo che matrix inoumbwa nemavector iwayo. Chigadzirwa che cross chinopihwa nefomura:
\[ \mathbf{F} \nguva \mathbf{G} = \kutanga{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \magumo{vmatrix} \]
Izvi zvinogona kuverengerwa nenzira inotevera:
\[
\mathbf{F} \nguva \mathbf{G} = \mathbf{i} \kutanga{vmatrix} 3 & 4 \\ -1 & 2 \end{vmatrix} - \mathbf{j} \kutanga{vmatrix} 2 & 4 \\ 1 & 2 \magumo \{vmatrix} +k \v 3 \matrix &{vmatrix} \v 3 & -1 \kupera{vmatrix}
\]
Kuverenga chinongedzo che submatrix yega yega:
\[
= \mathbf{i} (3 \cdot 2 – 4 \cdot -1) – \mathbf{j} (2 \cdot 2 – 4 \cdot 1) + \mathbf{k} (2 \cdot -1 – 3 \cdot 1)
\]
\[
= \mathbf{i} (6 + 4) – \mathbf{j} (4 – 4) + \mathbf{k} (-2 – 3)
\]
\[
= \mathbf{i} (10) – \mathbf{j} (0) + \mathbf{k} (-5)
\]
\[
= \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}
\]
Saka, chibereko chakasanganiswa cheF naG ndeichi:
\[ \mathbf{F} \times \mathbf{G} = \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix} \]
5. Kuziva Angle iri pakati pemaVector maviri
Muenzaniso Mubvunzo 5
Zvichipiwa mavector maviri H naI:
\[ \mathbf{H} = \begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{I} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} \]
Sarudza kona iri pakati pemavector maviri aya.
Kukurukurirana
Kona \(\theta\) iri pakati pemavekitari maviri inogona kuwanikwa nekushandisa hukama huripo pakati pechigadzirwa chedoti nehukuru hwemavekitari maviri:
\[ \mathbf{H} \cdot \mathbf{I} = \| \mathbf{H} \| \| \mathbf{I} \| \cos \theta \]
Kutanga, verenga chigadzirwa chedot \( \mathbf{H} \cdot \mathbf{I} \):
\[ \mathbf{H} \cdot \mathbf{I} = 6 \cdot 1 + 2 \cdot 4 + 3 \cdot (-2) = 6 + 8 – 6 = 8 \]
Tevere, verenga hukuru hwemavector ese ari maviri:
\[ \| \mathbf{H} \| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]
\[ \| \mathbf{I} \| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \]
Wobva watsiva izvi zvinhu mufomura yekona:
\[ \cos \theta = \frac{\mathbf{H} \cdot \mathbf{I}}{\| \mathbf{H} \| \| \mathbf{I} \|} = \frac{8}{7\sqrt{21}} \]
\[ \theta = \cos^{-1} \left( \frac{8}{7\sqrt{21}} \right) \]
Pakupedzisira, tinogona kushandisa karukureta kuti tiwane kukosha kwekona:
\[ \theta \approx 73,4^\circ \]
Mhedziso
Pfungwa yekushanda kwevector inokosha mumasvomhu nesainzi. Chinyorwa chino chinokurukura matambudziko akati wandei nemhinduro dzawo, kubva pakuwedzera nekubvisa vector, kuwanda kwescalar, dot product, cross product, uye kuona angle iri pakati pevector mbiri. Nekushandisa mienzaniso iyi, tinotarisira kuwedzera kunzwisisa kwako mashandiro evector uye kukubatsira kugadzirisa matambudziko ane chekuita nevectors mumamiriro akasiyana-siyana.
