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Question: Your directions on a scavenger hunt map say to walk 33 m east, then 41 m south, then 29 m northwest. The positive x direction is the direction to the east and the positive y direction is the direction to the north. Part A What is your displacement in polar coordinates?
In order to solve this task, first it would be helpful to convert everything into Cartesian coordinates, and then we can convert the resultant displacement into polar coordinates. Step by step: 1) The first displacement stands for 33 m to the east, which in Cartesian coordinates could be expressed as (+33 m, 0 m). 2) The second displacement of 41 m south translates into (0 m, -41 m). 3) The last part is a little trickier as it's not aligned with the axis. We have 29 m to the northwest, which is at a 45 degree angle. For this, we'll have to decompose this displacement into its x (westward) and y (northward) components. Because we're dealing with a 45-degree angle, both components will be equal in magnitude due to the properties of 45-45-90 right triangle. So, the third displacement decomposes to -29/sqrt(2) m in the x direction, and +29/sqrt(2) m in the y direction. 4) Now, add up all these displacements in both x and y directions separately. The x (east-west) direction: +33 m - 29/sqrt(2) m The y (north-south) direction: +29/sqrt(2) m - 41 m 5) Calculate these to find the final displacement in Cartesian coordinates. 6) The displacement in polar coordinates can be found by calculating the magnitude (r) and the angle (θ) using the Cartesian coordinates. The magnitude r is given by sqrt(x^2 + y^2) and the angle θ is found from the arctan(y/x). Remember that since we are working in the 4th quadrant (because our y is negative while x is positive), we have to add 360 degrees to our angle if it is negative or subtract 180 if it is positive in order to get the actual direction from the positive x-axis. This would be a simple way to break down your actual query if you require additional help in calculations or any step feel free to ask.
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