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 utatuzi wa swali la Sita: Swali la Sita a) Kutokana na vekta a=3i+5j, b=7i+3j, na c=-3i+2j, tafuta r= a+b-c, ukubwa wake |r| na mwelekeo wake. Step 1: Tafuta vekta r. r = a + b - c r = (3i + 5j) + (7i + 3j) - (-3i + 2j) r = (3+7-(-3))i + (5+3-2)j r = (3+7+3)i + (8-2)j r = 13i + 6j Step 2: Tafuta ukubwa (magnitude) wa r, |r|. |r| = sqrt(13^2 + 6^2) |r| = sqrt(169 + 36) |r| = sqrt(205) Step 3: Tafuta mwelekeo (direction) wa r. Mwelekeo hutolewa na pembe na mhimili chanya wa x. = (6)/(13) = ((6)/(13)) ≈ 24.77^ r = 13i + 6j, |r| = sqrt(205), mwelekeo ≈ 24.77^ b) Kutokana na vekta za nafasi za wanafunzi watatu OA=6i+2j, OB=3i+12j, na OC=7i+5j, tathmini yafuatayo: i) AC Step 1: Tumia formula AC = OC - OA. AC = (7i + 5j) - (6i + 2j) AC = (7-6)i + (5-2)j AC = 1i + 3j AC = i + 3j ii) |BC| Step 1: Tafuta vekta BC. BC = OC - OB BC = (7i + 5j) - (3i + 12j) BC = (7-3)i + (5-12)j BC = 4i - 7j Step 2: Tafuta ukubwa wa BC. |BC| = sqrt(4^2 + (-7)^2) |BC| = sqrt(16 + 49) |BC| = sqrt(65) |BC| = sqrt(65) iii) AB + 2AC Step 1: Tafuta vekta AB. AB = OB - OA AB = (3i + 12j) - (6i + 2j) AB = (3-6)i + (12-2)j AB = -3i + 10j Step 2: Tafuta 2AC. Kutoka sehemu (i), AC = i + 3j. 2AC = 2(i + 3j) 2AC = 2i + 6j Step 3: Tafuta jumla ya vekta. AB + 2AC = (-3i + 10j) + (2i + 6j) AB + 2AC = (-3+2)i + (10+6)j AB + 2AC = -1i + 16j AB + 2AC = -i + 16j Tuma swali linalofuata 📸