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) Express in terms of a and b. i) AB Tunajua OA = a na OB = b. Kwa kutumia sheria ya pembetatu ya vekta: AB = AO + OB Kwa kuwa AO = -OA = -a: AB = -a + b AB = b - a ii) OT T iko kwenye AB na inagawanya AB kwa uwiano wa 2:3 (AT:TB = 2:3). Kwa kutumia fomula ya sehemu ya vekta: OT = 3OA + 2OB2+3 OT = 3a + 2b5 OT = (3)/(5)a + (2)/(5)b iii) BP P inagawanya OA kwa uwiano wa 2:3 (OP:PA = 2:3). Hivyo, OP = (2)/(5)OA = (2)/(5)a. Kwa kutumia sheria ya pembetatu ya vekta: BP = BO + OP Kwa kuwa BO = -OB = -b: BP = -b + (2)/(5)a BP = (2)/(5)a - b b) Given that OX = kOT and BX = hBP. Express OX in two different ways. Hence find the values of the scalars k and h. Step 1: Andika OX kwa njia ya kwanza kwa kutumia OX = kOT. Tumia matokeo kutoka a) ii): OX = k((3)/(5)a + (2)/(5)b) OX = (3k)/(5)a + (2k)/(5)b (Equation 1) Step 2: Andika OX kwa njia ya pili kwa kutumia BX = hBP. Tunajua OX = OB + BX. Badilisha BX = hBP na OB = b: OX = b + hBP Tumia matokeo kutoka a) iii): OX = b + h((2)/(5)a - b) OX = b + (2h)/(5)a - hb OX = (2h)/(5)a + (1-h)b (Equation 2) Step 3: Linganisha semi mbili za OX na utatue kwa k na h. Kutoka Equation 1 na Equation 2: (3k)/(5)a + (2k)/(5)b = (2h)/(5)a + (1-h)b Kwa kuwa a na b si sambamba, tunaweza kulinganisha vizio vya a na b: Kulinganisha vizio vya a: (3k)/(5) = (2h)/(5) 3k = 2h (Equation 3) Kulinganisha vizio vya b: (2k)/(5) = 1-h 2k = 5(1-h) 2k = 5 - 5h (Equation 4) Step 4: Tatua mfumo wa milinganyo. Kutoka Equation 3, h = (3k)/(2). Badilisha h katika Equation 4: 2k = 5 - 5((3k)/(2)) 2k = 5 - (15k)/(2) Zidisha pande zote mbili kwa 2 ili kuondoa sehemu: 4k = 10 - 15k 4k + 15k = 10 19k = 10 k = (10)/(19) Sasa pata h: h = (3)/(2) × (10)/(19) h = (3 × 5)/(19) h = (15)/(19) Semi mbili za OX ni (3k)/(5)a + (2k)/(5)b na (2h)/(5)a + (1-h)b. Thamani za scalars ni k = (10)/(19) na h = (15)/(19). 4 down, 1 left today. Keep it up!