Su'aalo Tusaale ah oo Ka Hadlaya Vektors-ka Xun ama Vektors-ka Ka Soo Horjeeda
Xisaabta, gaar ahaan fiisikiska ama joomatari falanqayneed, fikradda vectors-ku waxay door muhiim ah ka ciyaartaa. Vectors-ku waxaa badanaa loo isticmaalaa inay matalaan tirooyin leh jiho iyo baaxad labadaba, sida xawaaraha, xoogga, iyo barokaca. Marka laga hadlayo vectors-ka, waxaan inta badan la kulannaa ereyada "vector taban" ama "vector ka soo horjeeda." Maqaalkani wuxuu si qoto dheer u sharxi doonaa fikraddan wuxuuna bixin doonaa tusaalooyin iyo xalal si loo fududeeyo fahamka.
Qeexitaanka Vektorka Taban
Vektor taban, ama vektor ka soo horjeeda, waa vektor leh jiho ka soo horjeeda laakiin leh cabbir la mid ah vektorkii asalka ahaa. Haddii aan haysanno vektor \(\mathbf{a}\), markaa vektor taban ee \(\mathbf{a}\), oo badanaa loo tilmaamo \(-\mathbf{a}\), wuxuu leeyahay jiho ka soo horjeeda iyo cabbir la mid ah kan \(\mathbf{a}\). Haddii \(\mathbf{a}\) lagu matalo qaabka qaybta sida \((a_x, a_y)\), markaa vektor taban waa \((-a_x, -a_y)\).
Tilmaamaha iyo Matalaadda Vektorka
Ka soo qaad in vektor \(\mathbf{a}\) lagu matalo qaabka qaybaha sida:
\[ \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} \]
halkaas oo \(\mathbf{i}\) iyo \(\mathbf{j}\) ay yihiin vectors cutubyo ku jira jihooyinka x- iyo y, siday u kala horreeyaan. Kadib, vector-ka taban \(\mathbf{a}\) ama \(-\mathbf{a}\) waxaa lagu matali karaa sidan:
\[ -\mathbf{a} = -a_x \mathbf{i} – a_y \mathbf{j} \]
Astaamaha Vektorrada Xun
Qaar ka mid ah sifooyinka muhiimka ah ee falgalayaasha taban waxaa ka mid ah:
1. Ku darista Vektor-ka Asalka ah: Ku darista vektor leh vektorkiisa taban waxay soo saari doontaa vektor eber ah.
\[ \mathbf{a} + (-\mathbf{a}) = \mathbf{0} \]
2. Hawlgallada Isle'egta: Ku dhufashada vektorka -1 waxay soo saari doontaa vektorkeeda taban.
\[ -1 \cdot \mathbf{a} = -\mathbf{a} \]
Su'aalo iyo Doodo Tusaale ah
Si aan si fiican u fahanno fikradda vectors taban ama vectors ka soo horjeeda, aan ka shaqeyno masalooyinka tusaalaha ah ee soo socda:
Tusaale 1:
Ka soo qaad inuu jiro vektor \(\mathbf{a} = 3 \mathbf{i} – 4 \mathbf{j}\). Go'aami vektorka taban ee vektorka \(\mathbf{a}\).
Dood:
Waa la ogyahay:
\[ \mathbf{a} = 3 \mathbf{i} - 4 \mathbf{j} \]
Vektorka taban ee \(\mathbf{a}\) waa:
\[ -\mathbf{a} = -1 \cdot (3 \mathbf{i} – 4 \mathbf{j}) \]
\[ -\mathbf{a} = -3 \mathbf{i} + 4 \mathbf{j} \]
Markaa, fallaadhaha taban ee \(\mathbf{a}\) waa:
\[ -\mathbf{a} = -3 \mathbf{i} + 4 \mathbf{j} \]
Tusaale 2:
Waxaa jira laba qaybood oo kala ah \(\mathbf{b} = 6 \mathbf{i} +2 \mathbf{j}\) iyo \(\mathbf{c} = -1 \mathbf{i} + 7 \mathbf{j}\). Soo hel badeecada \(\mathbf{b} + (-\mathbf{c})\).
Dood:
Waa la ogyahay:
\[ \mathbf{b} = 6 \mathbf{i} + 2 \mathbf{j} \]
\[ \mathbf{c} = -1 \mathbf{i} + 7 \mathbf{j} \]
Vektorka taban ee \(\mathbf{c}\) waa:
\[ -\mathbf{c} = -1 \cdot (-1 \mathbf{i} + 7 \mathbf{j}) \]
\[ -\mathbf{c} = 1 \mathbf{i} - 7 \mathbf{j} \]
Hadda waxaan helnaa \(\mathbf{b} + (-\mathbf{c})\):
\[ \mathbf{b} + (-\mathbf{c}) = (6 \mathbf{i} + 2 \mathbf{j}) + (1 \mathbf{i} - 7 \mathbf{j}) \]
\[ \mathbf{b} + (-\mathbf{c}) = (6 + 1) \mathbf{i} + (2 – 7) \mathbf{j} \]
\[ \mathbf{b} + (-\mathbf{c}) = 7 \mathbf{i} – 5 \mathbf{j} \]
Markaa, natiijada \(\mathbf{b} + (-\mathbf{c})\) waa:
\[ 7 \mathbf{i} - 5 \mathbf{j} \]
Tusaale 3:
Waxa jira vector \(\mathbf{d} = a \mathbf{i} + b \mathbf{j}\), halkaasoo a iyo b ay yihiin tiro dhab ah. Haddi \(\mathbf{d} + \mathbf{e} = \mathbf{0}\), go'aami vector \(\mathbf{e}\).
Dood:
Waa la ogyahay:
\[ \mathbf{d} = a \mathbf{i} + b \mathbf{j} \]
\[ \mathbf{d} + \mathbf{e} = \mathbf{0} \]
Si aan u helno \(\mathbf{e}\), waxaan qori karnaa:
\[ \mathbf{e} = -\mathbf{d} \]
Markaa, vektorka \(\mathbf{e}\) waa vektorka taban ee \(\mathbf{d}\):
\[ \mathbf{e} = -\mathbf{d} = -a \mathbf{i} – b \mathbf{j} \]
Tusaale 4:
Marka la eego vector-ka \(\mathbf{f} = 5 \mathbf{i} + k \mathbf{j}\). Waxaa la og yahay in xididka taban ee \(\mathbf{f}\) uu yahay \(-5 \mathbf{i} - 8 \mathbf{j}\). Go'aami qiimaha k.
Dood:
Waa la ogyahay:
\[ \mathbf{f} = 5 \mathbf{i} + k \mathbf{j} \]
\[ -\mathbf{f} = -5 \mathbf{i} - 8 \mathbf{j} \]
Xiriirkan, waxaan ka dhisi karnaa isle'egyada qaybaha ee \(\mathbf{f}\) iyo \(-\mathbf{f}\). Qayb ahaan, vektorka \(\mathbf{f}\) iyo vektorkiisa taban waa inay lahaadaan xiriir boos oo isku mid ah oo leh calaamado iska soo horjeeda. Sidaas darteed:
Qaybaha \( \mathbf{i} \):
\[ -5 = -5 \]
Tani si toos ah ayay run u tahay.
Qaybta \( \mathbf{j} \):
\[ -k = -8 \]
\[ k = 8 \]
Markaa, qiimaha \( k \) waa 8.
Gabagabo
Fahmidda fikradda vektor taban, ama vektor ka soo horjeeda, waa lama huraan marka la baranayo vektorrada. Vektor ka soo horjeeda waa vektor ka soo horjeeda jihada vektorka asalka ah laakiin leh baaxad isku mid ah. Hawlgallada vektorrada, aqoonsashada iyo isticmaalka vektorrada taban waxay noqon kartaa mid aad waxtar u leh fududeynta dhibaatooyin badan, sida ku darista ama kala-goynta vektorrada. Iyadoo la adeegsanayo ku dhaqanka iyo fahamka sifooyinka aasaasiga ah ee vektorrada, fahamka fikraddan waxay noqon doontaa mid aad u dareen badan.
Waxaan rajeyneynaa in su'aalaha tusaalaha ah iyo doodda lagu soo bandhigay maqaalkan ay kaa caawin doonaan inaad si qoto dheer u fahamto vectors-ka taban, ama vectors-ka iska soo horjeeda. Sii wad ku celcelinta iyo sahaminta su'aalo badan si aad u noqoto mid aad ugu xeel dheer agabkan!
