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: Substitute the given vectors into the equation r = pm + qn. We are given m = 2 \\ 3 , n = 3 \\ 2 , and r = 1 \\ -5 . 1 \\ -5 = p 2 \\ 3 + q 3 \\ 2 Step 2: Perform scalar multiplication on the right side of the equation. 1 \\ -5 = 2p \\ 3p + 3q \\ 2q Step 3: Add the vectors on the right side. 1 \\ -5 = 2p + 3q \\ 3p + 2q Step 4: Equate the corresponding components to form a system of linear equations. From the top components: 2p + 3q = 1 (1) From the bottom components: 3p + 2q = -5 (2) Step 5: Solve the system of equations for p and q. Multiply equation (1) by 3 and equation (2) by 2 to eliminate p: 3 × (2p + 3q = 1) 6p + 9q = 3 (3) 2 × (3p + 2q = -5) 6p + 4q = -10 (4) Subtract equation (4) from equation (3): (6p + 9q) - (6p + 4q) = 3 - (-10) 5q = 13 q = (13)/(5) Substitute the value of q into equation (1): 2p + 3((13)/(5)) = 1 2p + (39)/(5) = 1 2p = 1 - (39)/(5) 2p = (5)/(5) - (39)/(5) 2p = -(34)/(5) p = (-34)/(5 × 2) p = -(34)/(10) p = -(17)/(5) The values of the scalars are p = -(17)/(5) and q = (13)/(5). The final answer is p = -(17)/(5), q = (13)/(5). Send me the next one 📸