To solve this problem using the triangle method, we need to break the hiker's displacements into their x (horizontal) and y (vertical) components. We'll sum the components and then use the Pythagorean theorem to find the resultant displacement and its direction.

Step 1: Break each displacement into components

For each displacement:

• The hiker’s initial walk is 3000 m at 35° north of west.
• The second displacement is 5500 m at 10° north of west.
• The third displacement is 1500 m due north.

1. First displacement (3000 m, 35° north of west):

• x-component: $3000 \times \cos(35^\circ)$ (westward, so negative)
• y-component: $3000 \times \sin(35^\circ)$ (northward, so positive)

2. Second displacement (5500 m, 10° north of west):

• x-component: $5500 \times \cos(10^\circ)$ (westward, so negative)
• y-component: $5500 \times \sin(10^\circ)$ (northward, so positive)

3. Third displacement (1500 m due north):

• x-component: 0 (since it’s due north)
• y-component: 1500 (positive, since it's purely northward)

Step 2: Sum the components

• The total x-component of displacement is the sum of all x-components.
• The total y-component of displacement is the sum of all y-components.

Step 3: Calculate the resultant displacement and direction

The resultant displacement is the magnitude of the vector formed by the summed x and y components, calculated using the Pythagorean theorem:

$\text{Resultant displacement} = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2}$

The direction (θ) of the displacement, relative to west, can be found using the inverse tangent function:

$\theta = \tan^{-1}\left(\frac{y_{\text{total}}}{|x_{\text{total}}|}\right)$

Let me calculate this now.The hiker's total displacement is approximately 8912.66 meters.

The direction of the resultant displacement is about 27.94° north of west.

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. How does breaking vectors into components help solve displacement problems?
2. What is the significance of using the Pythagorean theorem for vector sums?
3. Why is trigonometry essential in calculating direction in two-dimensional motion?
4. Can you apply this method for displacements at angles other than north and west?
5. How would the result change if the third displacement was not due north?

Tip: Always keep track of directions carefully when breaking vectors into components, as the sign (positive or negative) is crucial for accuracy.