Orville walks 320 m due east. He then continues walking along a straight line, but in a different direction, and stops \(200 \mathrm{m}\) northeast of his starting point. How far did he walk during the second portion of the trip and in what direction?

To solve this problem, we can represent Orville's movements as vectors. Let A be his starting point, B be the point where he changes direction, and C be his final destination (200 meters northeast of point A). We can represent the first movement as the vector AB and the second movement as the vector BC. First movement (AB): 320 m due east Second movement (BC): unknown distance and direction

Set up a coordinate system, with the origin at point A, the x-axis pointing east, and the y-axis pointing north. In this coordinate system, the coordinates of points A, B, and C can be written as: A = (0,0) B = (320,0) C = (x, y) Now, we know that point C is 200 meters northeast of point A; thus, the displacement AC can be represented as: AC = (200 cos(45°), 200 sin(45°)) = (100√2, 100√2)

Using vector addition, we can relate the vectors AC, AB, and BC: AC = AB + BC (100√2, 100√2) = (320, 0) + BC We can now find the coordinates of point C by solving for BC: BC = (100√2, 100√2) - (320, 0) BC = (100√2 - 320, 100√2)

To find the distance and direction of the second movement (BC), we need to calculate the magnitude and angle of vector BC. The magnitude can be calculated using the Pythagorean theorem: \(Distance = |BC| = \sqrt{(100\sqrt{2} - 320)^2 + (100\sqrt{2})^2}\) \(Distance ≈ 324.50 m\) The direction can be found using the arctangent function: \(Direction = tan^{-1}\left(\frac{y}{x}\right) = tan^{-1}\left(\frac{100\sqrt{2}}{100\sqrt{2} - 320}\right)\) \(Direction ≈ 53.31°\) Since we are working in a coordinate system where the x-axis points east and the y-axis points north, this angle represents the direction measured counterclockwise from east. So, the second portion of the trip is approximately 324.50 meters long and in a direction approximately 53.31° counterclockwise from east.