Su'aalo Tusaale ah iyo Doodda Kala-goynta Vektor-ka
Pendahuluan
Xisaabta iyo fiisigiska, vektorrada waa fikrad aasaasi ah oo loo isticmaalo in lagu sharaxo ifafaale badan oo dabiici ah iyo injineernimo. Vekto waa tiro leh baaxad iyo jiha labadaba. Tusaalooyin muhiim ah oo vektorrada ah waa barokaca, xawaaraha, dardargelinta, iyo xoogga. Maqaalkan, waxaan ka hadli doonnaa kala-goynta vektorrada, inkastoo mowduucan inta badan lagu nuuxnuuxsado macnaha guud ee isku-darka vektorrada.
Kala-goynta Vektorka waa hawlgal aasaasi ah oo muhiim u ah falanqaynta vektorka. Si aan si qoto dheer ugu dhexgalno fikraddan, aan dib u eegno tusaalooyin dhibaatooyin ah iyo doodo la xiriira kala-goynta vektorka.
Kala-goynta Vektorka
Kala-goynta vektorka {\displaystyle \mathbf{A} – \mathbf{B}} waxaa lagu qeexaa hawlgalka lagu darayo vektorka {\displaystyle \mathbf{A}} oo leh vektorka {\displaystyle -\mathbf{B}}, halkaas oo {\displaystyle -\mathbf{B}} uu yahay vektor leh cabbir la mid ah kan {\displaystyle \mathbf{B}} laakiin leh jiho ka soo horjeeda. Xisaab ahaan, tan waxaa loo qori karaa sidan:
{\displaystyle \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})}
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad: Kala-goynta Vektorrada Laba-cabbir ah
Ka soo qaad in ay jiraan laba vectors oo ku jira isku-duwayaasha Cartesian:
{\displaystyle \mathbf{A} = (4, 3)} iyo {\displaystyle \mathbf{B} = (1, 2)}. Xisaabi {\displaystyle \mathbf{A} – \mathbf{B}}.
Dood:
Tallaabada ugu horreysa waa in la helo vector-ka taban ee {\displaystyle \mathbf{B}}, kuwaas oo kala ah:
{\displaystyle -\mathbf{B} = (-1, -2)}
Marka xigta, ku dar vector {\displaystyle \mathbf{A}} oo leh {\displaystyle -\mathbf{B}}:
{\ displaystyle \mathbf{A} - \mathbf{B} = (4, 3) + (-1, -2)}
Samee isku darka vector-ka adigoo ku daraya qayb kasta oo x iyo y ah:
{\ displaystyle \mathbf{A} - \mathbf{B} = (4 + (-1), 3 + (-2))}
{\ displaystyle \mathbf{A} - \mathbf{B} = (3, 1)}
Markaa, natiijada ka soo baxda kala-goynta vektorrada {\displaystyle \mathbf{A} – \mathbf{B}} waa vektorrada (3, 1).
Su'aal 2: Kala-goynta Vektorrada Saddex-cabbir ah
Marka la eego laba vector oo ku jira isku-duwayaasha saddex-geesoodka ah:
{\displaystyle \mathbf{P} = (2, -4, 6)} iyo {\displaystyle \mathbf{Q} = (-3, 5, 7)}. Xisaabi {\displaystyle \mathbf{P} – \mathbf{Q}}.
Dood:
Tallaabada ugu horreysa waa in la helo vector-ka taban ee {\displaystyle \mathbf{Q}}:
{\displaystyle -\mathbf{Q} = (3, -5, -7)}
Marka xigta, ku dar vector {\displaystyle \mathbf{P}} oo leh {\displaystyle -\mathbf{Q}}:
{\ displaystyle \mathbf{P} - \mathbf{Q} = (2, -4, 6) + (3, -5, -7)}
Samee isku darka vector-ka adigoo ku daraya qayb kasta oo x, y, iyo z ah:
{\displaystyle \mathbf{P} - \mathbf{Q} = (2 + 3, -4 + (-5), 6 + (-7))}
{\displaystyle \mathbf{P} - \mathbf{Q} = (5, -9, -1)}
Markaa, natiijada ka soo baxda kala-goynta vektorrada {\displaystyle \mathbf{P} – \mathbf{Q}} waa vektorrada (5, -9, -1).
Su'aal 3: Kala-goynta Vektor-ka ee Diyaaradda Isku-dhafan
Ka soo qaad inay jiraan laba fallaadho oo ay matalaan tirooyin isku dhafan:
{\displaystyle \mathbf{M} = 3 + 4i} iyo {\displaystyle \mathbf{N} = 1 + 2i}. Xisaabi {\displaystyle \mathbf{M} – \mathbf{N}}.
Dood:
Tallaabada ugu horreysa waa in la helo vector-ka taban ee {\displaystyle \mathbf{N}}:
{\ displaystyle -\mathbf{N} = -1 - 2i}
Marka xigta, ku dar vector {\displaystyle \mathbf{M}} oo leh {\displaystyle -\mathbf{N}}:
{\displaystyle \mathbf{M} - \mathbf{N} = (3 + 4i) + (-1 - 2i)}
Samee isku darka vector-ka adigoo ku daraya qayb kasta oo dhab ah iyo mid male-awaal ah:
{\displaystyle \mathbf{M} - \mathbf{N} = (3 + (-1)) + (4i + (-2i))}
{\ displaystyle \mathbf{M} - \mathbf{N} = 2 + 2i}
Markaa, natiijada ka soo baxda kala-goynta vector-yada {\displaystyle \mathbf{M} – \mathbf{N}} waa tirada isku dhafan ee 2 + 2i.
Su'aal 4: Kala-goynta Vektor-ka ee Nidaamka Isku-dubaridka Polar-ka
Ka soo qaad in ay jiraan laba vectors oo ku jira isku-duwayaasha cidhifka:
{\displaystyle \mathbf{U}} waxay leedahay cabbir dhan 5 iyo xagal dhan 30°,
iyo {\displaystyle \mathbf{V}} waxay leedahay cabbir dhan 3 iyo xagal dhan 150°.
Xisaabi {\displaystyle \mathbf{U} – \mathbf{V}}.
Dood:
Tallaabada ugu horreysa waa in vektorrada {\displaystyle \mathbf{U}} iyo {\displaystyle \mathbf{V}} loo beddelo isku-duwayaasha Cartesian.
Loogu talagalay {\displaystyle \mathbf{U}}:
{\displaystyle U_x = 5 \cos(30^\circ) = 5 \left(\frac{\sqrt{3}}{2}\right) = 5 \cdot 0.866 = 4.33}
{\displaystyle U_y = 5 \sin(30^\circ) = 5 \left(\frac{1}{2}\right) = 5 \cdot 0.5 = 2.5}
Markaa {\displaystyle \mathbf{U}} ee Cartesian waa (4.33, 2.5).
Loogu talagalay {\displaystyle \mathbf{V}}:
{\displaystyle V_x = 3 \cos(150^\circ) = 3 \left(\frac{-\sqrt{3}}{2}\right) = 3 \cdot (-0.866) = -2.598}
{\displaystyle V_y = 3 \sin(150^\circ) = 3 \left(\frac{1}{2}\right) = 3 \cdot 0.5 = 1.5}
Markaa {\displaystyle \mathbf{V}} ee Cartesian waa (-2.598, 1.5).
Tallaabada xigta, xisaabi kala-goynta vector-ka ee Cartesian:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33, 2.5) – (-2.598, 1.5)}
Taas oo macnaheedu yahay marka lagu daro vektorka taban:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33 + 2.598, 2.5 – 1.5)}
{\displaystyle \mathbf{U} – \mathbf{V} = (6.928, 1)}
Markaa, natiijada ka soo baxda kala-goynta vektor-ka {\displaystyle \mathbf{U} – \mathbf{V}} ee isku-duwayaasha Cartesian waa (6.928, 1).
Gabagabo
Kala-goynta Vektorku waa hawlgal xisaabeed oo muhiim ah oo ku saabsan dhinacyo badan oo adeegsada falanqaynta vektorka. Hadday tahay nidaamyada isku-dubaridka laba-geesoodka ah, saddex-geesoodka ah, isku-dhafan, ama kuwa cidhifka ah, mabda'a aasaasiga ah wuxuu ahaanayaa isku mid: ku darista hal vektor taban mid kale. Tusaalooyinka kor ku xusan waxay muujinayaan siyaabo kala duwan oo loogu dabaqi karo hawlgalkan xaalado kala duwan, taasoo naga caawinaysa inaan si qoto dheer oo ficil ah u fahanno fikradda.
