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 how to express QS in terms of p and r: Step 1: Express OQ in terms of p and r. Given that PQ:QR = 1:2, point Q divides the line segment PR in the ratio 1:2. Using the section formula for position vectors: OQ = 2OP + 1OR1+2 Substitute OP = p and OR = r: OQ = 2p + r3 Step 2: Express OS in terms of p and r. Given that PS = 3PR, this means the vector PS is 3 times the vector PR. PS = 3PR We know that PS = OS - OP and PR = OR - OP. OS - OP = 3(OR - OP) Substitute OP = p and OR = r: OS - p = 3(r - p) OS - p = 3r - 3p OS = 3r - 3p + p OS = 3r - 2p Step 3: Calculate QS. To find QS, we use the relationship QS = OS - OQ. Substitute the expressions for OS and OQ from Step 1 and Step 2: QS = (3r - 2p) - (2p + r3) QS = 3r - 2p - (2)/(3)p - (1)/(3)r Combine the terms with p and r: QS = (3 - (1)/(3))r + (-2 - (2)/(3))p QS = ((9)/(3) - (1)/(3))r + (-(6)/(3) - (2)/(3))p QS = (8)/(3)r - (8)/(3)p QS = (8)/(3)(r - p) That's 2 down. 3 left today — send the next one.