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 find the resultant displacement for question 1: Step 1: Represent the displacements as vectors. The first displacement is 12 km south. Let's represent South as the negative y-direction. d_1 = (0, -12) km The second displacement is 5 km east. Let's represent East as the positive x-direction. d_2 = (5, 0) km Step 2: Calculate the resultant vector by adding the component vectors. The resultant displacement R is the sum of d_1 and d_2. R = d_1 + d_2 = (0+5, -12+0) = (5, -12) km This means the resultant displacement is 5 km East and 12 km South. Step 3: Calculate the magnitude of the resultant displacement. The magnitude of the resultant vector is found using the Pythagorean theorem: |R| = sqrt(R_x^2 + R_y^2) |R| = sqrt((5)^2 + (-12)^2) |R| = sqrt(25 + 144) |R| = sqrt(169) |R| = 13 km Step 4: Calculate the direction of the resultant displacement. The resultant vector is in the fourth quadrant (East and South). Let be the angle measured from the East direction towards the South. = oppositeadjacent = (|R_y|)/(|R_x|) = (12)/(5) = ((12)/(5)) ≈ 67.38^ The direction is 67.38^ South of East. The magnitude of the resultant displacement is 13 km and the direction is 67.38^ South of East. 3 done, 2 left today. You're making progress.