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
Here's the solution for question 10. Given the vectors OA = (a-1)i + 2j and OB = 3ai + 6j. a) Find the vector AB in terms of a. Step 1: Use the vector subtraction formula. The vector AB can be found by subtracting vector OA from vector OB: AB = OB - OA Step 2: Substitute the given expressions for OA and OB. AB = (3ai + 6j) - ((a-1)i + 2j) Step 3: Group the i components and j components. AB = (3a - (a-1))i + (6 - 2)j Step 4: Simplify the expression. AB = (3a - a + 1)i + (4)j AB = (2a + 1)i + 4j The vector AB in terms of a is (2a+1)i + 4j. b) Find the value of a given that AB = 4i + 2j. Step 1: Set the expression for AB from part (a) equal to the given vector. We have AB = (2a+1)i + 4j and we are given AB = 4i + 2j. Therefore: (2a+1)i + 4j = 4i + 2j Step 2: Equate the corresponding components. For the i components: 2a+1 = 4 For the j components: 4 = 2 Step 3: Analyze the equations. The equation from the j components, 4=2, is a contradiction. This indicates there might be an error in the problem statement as it's impossible for the j components to be equal. However, if we assume there's a typo in the given AB for part (b) and proceed with the i components to find a: 2a+1 = 4 2a = 4 - 1 2a = 3 a = (3)/(2) Given the contradiction in the j components, the problem as stated has no solution for a that satisfies both components simultaneously. If we are forced to choose one component, the i component gives a = (3)/(2). Assuming the question intended for the j component of AB to be 4j in part (b) as well, then the value of a would be derived solely from the i components. The value of a is (3)/(2). 3 done, 2 left today. You're making progress.