Su'aalo Tusaale ah oo Ka Hadlaya Vektors-ka Isku-dhafka ah ee Nidaamka Isku-xirka Cartesian
Pendahuluan
Xisaabta, vektor waa hay'ad leh baaxad iyo jiha labadaba. Vektoradu waxay leeyihiin codsiyo dhinacyo kala duwan sida fiisigiska, injineernimada, iyo sayniska kombiyuutarka. Maqaalkan, waxaan ka hadli doonnaa fikradda vektorrada u dhigma ee nidaamka isku-duwidda Cartesian waxaanan soo bandhigi doonnaa tusaalooyin iyo xalal. Fahmidda vektorrada u dhigma waa muhiim codsiyada kala duwan, oo ay ku jiraan farsamada iyo sawirada kombiyuutarka.
Aasaaska Vektors-ka ee Nidaamka Isku-xirka Cartesian
Nidaamka isku-dubaridka Cartesian waa nidaam laba-geesood ah oo leh dhidibyada X iyo Y oo is dul saaran. Nidaamkan, dhidibyada waxaa badanaa loo matalaa sidii lammaane la kala horraysiiyay (x, y), halkaas oo x iyo y ay yihiin qaybaha dhidibka ee ku teedsan dhidibyada X iyo Y, siday u kala horreeyaan.
Ka soo qaad inaan laba dhibcood ku leenahay nidaamka isku-duwidda Cartesian, \(A(x_1, y_1)\) iyo \(B(x_2, y_2)\). Vektor-ka isku xira labadan dhibcood waxaa loo tilmaami karaa \( \vec{AB} = (x_2 – x_1, y_2 – y_1) \).
Vektorrada U dhigma
Laba vektor ayaa la sheegaa inay isku mid yihiin haddii ay leeyihiin baaxad iyo jiha isku mid ah. Xisaab ahaan, laba vektor \( \vec{u} = (u_1, u_2) \) iyo \( \vec{v} = (v_1, v_2) \) waa isku mid haddii iyo haddii:
\[
\vec{u} = \vec{v} \quad \text{or} \quad (u_1 = v_1 \text{ iyo } u_2 = v_2)
\]
Taas macnaheedu waa in qaybaha u dhigma ee labada vector ay isku mid yihiin.
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad: Go'aaminta Vektorrada Isku-midka ah
Saddex dhibcood ayaa lagu bixiyay nidaamka isku-duwidda Kartesian: \( A(2, 3) \), \( B(5, 7) \), iyo \( C(7, -1) \). Go'aami in vektorka \( \vec{AB} \) uu la mid yahay vektorka \( \vec{AC} \).
Dood:
– Go'aami vektor \( \vec{AB} \):
\[
\vec{AB} = (5 - 2, 7 - 3) = (3, 4)
\]
– Go'aami vektor \( \vec{AC} \):
\[
\vec{AC} = (7 – 2, -1 – 3) = (5, -4)
\]
Ka dib markaan xisaabinno qaybaha vektor kasta, waxaan aragnaa in \( \vec{AB} = (3, 4) \) iyo \( \vec{AC} = (5, -4) \). Maadaama \( (3, 4) \neq (5, -4) \), vektor \( \vec{AB} \) uusan la mid ahayn vektor \( \vec{AC} \).
Su'aal 2aad: Dhisidda Vektors-ka Isku-midka ah
Go'aami barta \( D \) si vektorka \( \vec{AB} = \vec{CD} \) oo leh dhibic \( C(4, -2) \), dhibic \( B(8, 3) \), iyo \( A(2, 1) \).
Dood:
– Go'aami vektor \( \vec{AB} \):
\[
\vec{AB} = (8 - 2, 3 - 1) = (6, 2)
\]
Maadaama \( \vec{CD} \) ay tahay inay la mid noqoto \( \vec{AB} \), markaa:
\[
\vec{CD} = \vec{AB} = (6, 2)
\]
– Ka soo qaad \( D(x, y) \). Kadib \( \vec{CD} = (x – 4, y + 2) \). Halkan waxaan ka helnaa:
\[
(x – 4, y + 2) = (6, 2)
\]
Marka la barbardhigo qaybaha ku habboon, waxaan helnaa:
\[
x – 4 = 6 \quad \Midigtaro \quad x = 10
\]
\[
y + 2 = 2 \quad \Midigta fallaadha \quad y = 0
\]
Markaa, qodobka \( D \) waa \( (10, 0) \).
Su'aal 3: Caddeyn leh Cabbirka Vektor-ka
Caddee in vektorrada \( \vec{PQ} \) iyo \( \vec{RS} \) ay isku mid yihiin, marka la eego \( P(1, 2) \), \( Q(4, 6) \), \( R(-3, -7) \), iyo \( S(0, -3) \).
Dood:
– Go'aami vektor \( \vec{PQ} \):
\[
\vec{PQ} = (4 – 1, 6 – 2) = (3, 4)
\]
– Qeex vektorka \( \vec{RS} \):
\[
\vec{RS} = (0 – (-3), -3 – (-7)) = (3, 4)
\]
Natiijooyinka xisaabinta, waxaan ku aragnaa in \( \vec{PQ} = (3, 4) \) iyo \( \vec{RS} = (3, 4) \). Maadaama labada vector ay leeyihiin qaybo isku mid ah, \( \vec{PQ} \) waxay u dhigantaa \( \vec{RS} \).
Adeegsiga Vektorrada U dhigma
Vektorrada u dhigma ayaa si joogto ah loogu isticmaalaa qaybaha kala duwan ee sayniska. Fiisigiska, waxaa loo isticmaalaa in lagu qeexo xoogagga ama barokaca leh baaxadda iyo jihada isku midka ah. Garaafyada kombiyuutarka, vektorrada waxaa loo isticmaalaa in si hufan loogu beddelo oo loogu hawlgeliyo walxaha garaafka.
Gabagabo
Fahmidda fikradda vektorrada u dhigma ee nidaamka isku-dubaridka Cartesian waa aasaas muhiim u ah xisaabta iyo adeegsigeeda ballaaran. Maqaalkani wuxuu ka hadlay sida loo go'aamiyo vektorrada u dhigma iyada oo loo marayo dhowr masalooyin tusaale ah iyo xalalkooda. Annagoo fahanayna oo adeegsanayna fikraddan, waxaan xallin karnaa dhibaatooyin kala duwan oo ku lug leh falanqaynta vektorrada ee dhinacyo badan oo sayniska ah.
Waxaan rajeyneynaa in dooddani ay kaa caawin doonto inaad fahamto fikradda vectors-ka u dhigma ee nidaamka isku-dubaridka Cartesian. Barasho wanaagsan, iyo nasiib wacan oo ku saabsan barashada vectors-ka!
