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
Step 1: Express OP in terms of a and b. Given that P is a point on OA such that OP = (1)/(3) OA. Since OA = a, we have: OP = (1)/(3) OA = (1)/(3) a Step 2: Express OQ in terms of a and b. Given that Q is a point on OB such that OQ = (1)/(2) OB. Since OB = b, we have: OQ = (1)/(2) OB = (1)/(2) b Step 3: Express OR in terms of a and b. Given that R is a point on AB such that AR = (1)/(3) AB. First, find the vector AB: AB = OB - OA = b - a Now, express AR: AR = (1)/(3) AB = (1)/(3) (b - a) Using vector addition, OR = OA + AR: OR = a + (1)/(3) (b - a) OR = a + (1)/(3) b - (1)/(3) a OR = (1 - (1)/(3)) a + (1)/(3) b = (2)/(3) a + (1)/(3) b Step 4: Calculate PR. Using the relation PR = OR - OP: PR = ((2)/(3) a + (1)/(3) b) - ((1)/(3) a) PR = (2)/(3) a - (1)/(3) a + (1)/(3) b PR = (1)/(3) a + (1)/(3) b Step 5: Calculate RQ. Using the relation RQ = OQ - OR: RQ = ((1)/(2) b) - ((2)/(3) a + (1)/(3) b) RQ = (1)/(2) b - (2)/(3) a - (1)/(3) b RQ = -(2)/(3) a + ((1)/(2) - (1)/(3)) b RQ = -(2)/(3) a + ((3)/(6) - (2)/(6)) b RQ = -(2)/(3) a + (1)/(6) b Step 6: Compare PR and RQ. From Step 4, PR = (1)/(3) a + (1)/(3) b. From Step 5, RQ = -(2)/(3) a + (1)/(6) b. Since the coefficients of a and b are not equal in both expressions, PR ≠ RQ. Therefore, the statement PR = RQ cannot be shown to be true under the given conditions. Send me the next one 📸