In wandering, a grizzly bear makes a displacement of 1586 m due west, followed by a displacement of 3289 m in a direction 30.0° north of west. What are the magnitude and the direction of the displacement needed for the bear to return to its starting point? Specify the direction relative to due east.

I believe you add the two vectors, but I am not sure whether to just add 1586+3289cos(30.0) or add something else as well?

psychics? poltergeists??

physics sorry

To determine the magnitude and direction of the displacement needed for the bear to return to its starting point, you need to add the two displacement vectors. Adding only the magnitudes of the two vectors, as you mentioned, would not give you the correct answer.

Let's break down the problem step-by-step:

1. Start with the first displacement of 1586 m due west.
- Since it is due west, the angle between this vector and the east direction is 90°.
- This vector's magnitude remains 1586 m.

2. Move on to the second displacement of 3289 m in a direction 30.0° north of west.
- To combine this vector with the previous one, we need to resolve it into its north and west components.
- The west component is given by 3289 m × cos(30.0°).
- west component = 3289 m × cos(30.0°) = 3289 m × 0.866 = 2850.974 m (rounded to four decimal places).
- The north component is given by 3289 m × sin(30.0°).
- north component = 3289 m × sin(30.0°) = 3289 m × 0.5 = 1644.5 m.

3. Now, add the north and west components separately with the respective components from the first displacement:
- For the west component: 1586 m + 2850.974 m = 4436.974 m (rounded to four decimal places).
- For the north component: 1644.5 m.
- Note that the north component remains the same since there is no other northward displacement.

4. Finally, to find the magnitude and direction of the displacement needed to return to the starting point, we can use the Pythagorean theorem and trigonometry.
- The magnitude of the displacement is given by: magnitude = √(north component² + west component²).
- magnitude = √((1644.5 m)² + (4436.974 m)²) ≈ 4775.78 m.
- The direction of the displacement relative to due east can be calculated using the arctan function:
- direction = arctan(north component / west component).
- direction = arctan(1644.5 m / 4436.974 m) ≈ 20.768°.

So, to return to its starting point, the bear needs a displacement with a magnitude of approximately 4775.78 m and a direction of about 20.768° north of east.

To find the magnitude and direction of the displacement needed for the bear to return to its starting point, you need to add the two vectors together. Here's how you can do it step by step:

1. Start by adding the two displacement vectors. The first displacement of 1586 m due west is represented as (-1586 m, 0°) since it is along the negative x-axis. The second displacement of 3289 m in a direction 30.0° north of west can be represented as (-3289 m, -30.0°).

2. To add the two vectors, you need to break them down into their x and y components. For the first vector, since it is entirely in the x-direction, the x-component is -1586 m and the y-component is 0 m. For the second vector, you need to find its x and y components using trigonometry. The x-component is -3289 m * cos(30.0°) and the y-component is -3289 m * sin(30.0°).

3. Now, you can add the x and y components separately. The resulting x-component is -1586 m + (-3289 m * cos(30.0°)) and the resulting y-component is 0 m + (-3289 m * sin(30.0°)).

4. Calculate the magnitude of the resultant vector using the Pythagorean theorem: magnitude = sqrt((x-component)^2 + (y-component)^2).

5. Finally, find the direction of the resultant vector relative to due east using trigonometry. Use the arctangent of (y-component / x-component) and convert the angle from radians to degrees.

By following these steps, you will be able to determine the magnitude and direction of the displacement needed for the bear to return to its starting point.