The problem on the screen asks for the total displacement of a hiker who moves in multiple directions.

To solve this, we will use vector addition. The hiker takes three steps, each involving movement at specific angles and distances. Here's a breakdown:

1. First step: 3000 meters at 35° north of west.
2. Second step: 5500 meters at 10° north of west.
3. Final step: 1500 meters due north.

We'll break each vector into components (x and y axes) using trigonometry, sum the components, and then calculate the magnitude of the resultant vector (displacement).

Let's denote:

• $\text{X-axis}$ as east-west (positive x towards east),
• $\text{Y-axis}$ as north-south (positive y towards north).

Step 1: Vector Components for Each Movement

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

   • $x_1 = 3000 \cos(180^\circ - 35^\circ) = -3000 \cos(35^\circ)$
   • $y_1 = 3000 \sin(35^\circ)$

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

   • $x_2 = 5500 \cos(180^\circ - 10^\circ) = -5500 \cos(10^\circ)$
   • $y_2 = 5500 \sin(10^\circ)$

3. Third movement (1500 m due north):

   • $x_3 = 0$
   • $y_3 = 1500$

Now let's compute these values.The total displacement of the hiker is calculated as follows:

• The total displacement vector's x-component is $-7873.90$ meters (west).
• The total displacement vector's y-component is $4175.79$ meters (north).

Using these, the total displacement is approximately $8912.66$ meters.

This is the straight-line distance from the starting point to the hiker's final position.

Would you like further details or explanations on any steps? Here are some follow-up questions that might expand your understanding:

1. How can we use trigonometry to break vectors into components?
2. What does the negative x-component signify in vector problems?
3. How do you calculate the direction of the resultant displacement?
4. Why is vector addition important in physics problems like this one?
5. What are other real-world examples where total displacement is useful?

Tip: In vector problems, breaking vectors into components simplifies calculations, especially when adding multiple vectors.