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 solve the problem: a) Find the coordinates of B and C. Step 1: Find the coordinates of B. We are given the coordinates of A(4, 7) and the vector BA = 5 \\ 3 . The vector BA is calculated as A - B. Let B = (x_B, y_B). 4 \\ 7 - x_B \\ y_B = 5 \\ 3 This gives us two equations: 4 - x_B = 5 x_B = 4 - 5 = -1 7 - y_B = 3 y_B = 7 - 3 = 4 So, the coordinates of B are (-1, 4). Step 2: Find the coordinates of C. We are given the coordinates of A(4, 7) and the vector AC = 4 \\ -3 . The vector AC is calculated as C - A. Let C = (x_C, y_C). x_C \\ y_C - 4 \\ 7 = 4 \\ -3 This gives us two equations: x_C - 4 = 4 x_C = 4 + 4 = 8 y_C - 7 = -3 y_C = 7 - 3 = 4 So, the coordinates of C are (8, 4). The coordinates are: B(-1, 4), C(8, 4) b) If M is the midpoint of the line BC, find AM. Step 3: Find the coordinates of M. M is the midpoint of BC. We use the midpoint formula: M = ( (x_1 + x_2)/(2), (y_1 + y_2)/(2) ). Using B(-1, 4) and C(8, 4): M = ( (-1 + 8)/(2), (4 + 4)/(2) ) = ( (7)/(2), (8)/(2) ) = (3.5, 4) So, the coordinates of M are (3.5, 4). Step 4: Find the vector AM. The vector AM is calculated as M - A. Using A(4, 7) and M(3.5, 4): AM = 3.5 \\ 4 - 4 \\ 7 = 3.5 - 4 \\ 4 - 7 = -0.5 \\ -3 The vector AM is: -0.5 \\ -3 That's 2 down. 3 left today — send the next one.