Determine the resultant displacement rising analytical method

d1 = 400 km, E
d2 = 200 km, 20 degrees W of S
d3 = 280 km, N
d4 = 350 km, 70 degrees E of S

Find the components of d2 and d4 using trig.

Then sum all E-W components (paying attention to direction). Then sum N-S components. Then use Pythagoean to get resultant.

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To determine the resultant displacement using analytical methods, follow these steps:

1. Assign a coordinate system: Choose a coordinate system that is convenient for calculating the displacements. In this case, let's choose a coordinate system where North is the positive y-axis and East is the positive x-axis.

2. Break down the displacements: Break down each displacement vector into its vertical (y-component) and horizontal (x-component) parts. To do this, you can use trigonometry.

- For d1 = 400 km, E: The displacement is purely horizontal (along the positive x-axis), so its x-component is 400 km and the y-component is 0 km.

- For d2 = 200 km, 20 degrees W of S: The displacement has both vertical and horizontal components. To find the components, you can use trigonometry. Since it is 20 degrees West of South, the angle with the positive x-axis is 70 degrees. Therefore, the x-component is -200 * cos(70°) km (negative because it is in the West direction), and the y-component is -200 * sin(70°) km (negative because it is in the South direction).

- For d3 = 280 km, N: The displacement is purely vertical (along the positive y-axis), so its x-component is 0 km and the y-component is 280 km.

- For d4 = 350 km, 70 degrees E of S: Similar to d2, the angle with the positive x-axis is 20 degrees (East is positive). Therefore, the x-component is 350 * cos(20°) km, and the y-component is -350 * sin(20°) km (negative because it is in the South direction).

3. Add the components: Add up all the x-components (horizontal direction) and add up all the y-components (vertical direction) to find the total x-component (Rx) and total y-component (Ry).

4. Find the resultant displacement: Use the Pythagorean theorem and trigonometry to find the magnitude (R) and direction (θ) of the resultant displacement vector.

- Magnitude: R = sqrt(Rx^2 + Ry^2)
- Direction: θ = tan^(-1)(Ry/Rx)

By following these steps, you can determine the resultant displacement using analytical methods.