This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Huu hapa ni ufumbuzi wa swali lako: a) Pata OC Step 1: Tumia sifa za parallelogram. Katika parallelogram OABC, vekta OC ni sawa na vekta AB. OC = AB OC = b Vekta OC ni b. b) Pata BN kwa kutumia a na b Step 1: Eleza ON. Pointi N iko kwenye OA na ON : NA = 1 : 3. Hii inamaanisha ON ni sehemu ya OA. ON = (1)/(1+3) OA = (1)/(4) OA ON = (1)/(4)a Step 2: Eleza OB. Tumia njia kutoka O kwenda B. OB = OA + AB OB = a + b Step 3: Pata BN. Tumia njia kutoka B kwenda N. BN = BO + ON Kumbuka BO = -OB. BN = -(a + b) + (1)/(4)a BN = -a - b + (1)/(4)a BN = (1 - (1)/(4))a - b BN = -(3)/(4)a - b c) Pata thamani ya h na k Step 1: Eleza AC. AC = AO + OC Kumbuka AO = -OA. AC = -a + b Step 2: Eleza OX kwa njia ya kwanza (kutumia h). Tunajua AX = hAC. OX = OA + AX OX = OA + hAC OX = a + h(-a + b) OX = a - ha + hb OX = (1-h)a + hb (Mlinganyo 1) Step 3: Eleza OX kwa njia ya pili (kutumia k). Tunajua BX = kBN. OX = OB + BX OX = OB + kBN Tulipata OB = a + b na BN = -(3)/(4)a - b. OX = (a + b) + k(-(3)/(4)a - b) OX = a + b - (3)/(4)ka - kb OX = (1 - (3)/(4)k)a + (1-k)b (Mlinganyo 2) Step 4: Linganisha vigawo vya a na b kutoka Mlinganyo 1 na Mlinganyo 2. Kwa a: 1-h = 1 - (3)/(4)k h = (3)/(4)k (Mlinganyo 3) Kwa b: h = 1-k (Mlinganyo 4) Step 5: Tatua milinganyo ya wakati mmoja kupata h na k. Badilisha Mlinganyo 4 kwenye Mlinganyo 3: 1-k = (3)/(4)k 1 = k + (3)/(4)k 1 = (4k+3k)/(4) 1 = (7)/(4)k k = (4)/(7) Sasa pata h kwa kutumia h = 1-k: h = 1 - (4)/(7) h = (7-4)/(7) h = (3)/(7) Thamani za h na k ni h = (3)/(7), k = (4)/(7). d) Eleza OX kwa kutumia a na b Step 1: Tumia Mlinganyo 1 na thamani ya h uliyopata. OX = (1-h)a + hb Badilisha h = (3)/(7): OX = (1-(3)/(7))a + (3)/(7)b OX = ((7-3)/(7))a + (3)/(7)b OX = (4)/(7)a + (3)/(7)b That's 2 down. 3 left today — send the next one.