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.
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Haya, twende tukasuluhishe swali hili la vekta. Taarifa tulizopewa: OA = a OB = b X iko kwenye OA na OX = (1)/(2)OA = (1)/(2)a. Y iko kwenye AB na BY:YA = 1:3. Mstari BX na OY hukutana kwenye C. a) Eleza kwa kutumia a na b: 1) BA Step 1: Tumia sheria ya pembetatu ya vekta. BA = BO + OA BA = -OB + OA BA = -b + a BA = a - b 2) OY Step 1: Tumia fomula ya sehemu ya vekta kwa pointi Y inayogawanya BA kwa uwiano 1:3. OY = 3OB + 1OA1+3 OY = 3b + a4 OY = (1)/(4)a + (3)/(4)b 3) BX Step 1: Tumia sheria ya pembetatu ya vekta. BX = BO + OX BX = -OB + (1)/(2)OA BX = -b + (1)/(2)a BX = (1)/(2)a - b b) Kutokana na OC = hOY na BC = kBX, tafuta thamani za h na k. Step 1: Eleza OC kwa njia mbili tofauti. Kutoka OC = hOY: OC = h((1)/(4)a + (3)/(4)b) OC = (h)/(4)a + (3h)/(4)b (Equation 1) Kutoka BC = kBX, tunaweza kuandika OC kama: OC = OB + BC OC = OB + kBX OC = b + k((1)/(2)a - b) OC = b + (k)/(2)a - kb OC = (k)/(2)a + (1-k)b (Equation 2) Step 2: Linganisha vigawo vya a na b kutoka Equation 1 na Equation 2. Kulinganisha vigawo vya a: (h)/(4) = (k)/(2) h = 2k (Equation 3) Kulinganisha vigawo vya b: (3h)/(4) = 1-k (Equation 4) Step 3: Tatua milinganyo ya wakati mmoja kupata h na k. Badilisha Equation 3 ndani ya Equation 4: (3(2k))/(4) = 1-k (6k)/(4) = 1-k (3k)/(2) = 1-k Kuzidisha pande zote mbili kwa 2: 3k = 2(1-k) 3k = 2 - 2k 3k + 2k = 2 5k = 2 k = (2)/(5) Step 4: Pata thamani ya h kwa kutumia Equation 3. h = 2k h = 2((2)/(5)) h = (4)/(5) 3 done, 2 left today. You're making progress.