To determine the resultant displacement of the truck, we need to break down its movement into vector components, sum them, and then draw a scale diagram to visualize the situation. Here’s how you can approach the problem:

Step-by-Step Process:

1. First Movement: 1 mile North

   • This is a vertical vector. It has no horizontal component, and its vertical component is $+1 \, \text{mile}$ (since it's in the positive y-direction).

2. Second Movement: 2 miles East

   • This is a horizontal vector. It has no vertical component, and its horizontal component is $+2 \, \text{miles}$ (since it's in the positive x-direction).

3. Third Movement: 3 miles Northwest

   • Northwest is at a 45° angle between North and West, so we need to break this movement into its horizontal and vertical components.
   • For a vector at 45°, the horizontal and vertical components are equal, given by: $\text{Horizontal Component} = -3 \cos(45^\circ) = -3 \times \frac{\sqrt{2}}{2} = -2.12 \, \text{miles}$ $\text{Vertical Component} = +3 \sin(45^\circ) = +3 \times \frac{\sqrt{2}}{2} = +2.12 \, \text{miles}$
   • The negative horizontal component is because the direction is West, and the positive vertical component is because the direction is North.

Summing the Components:

Now, sum the horizontal and vertical components to find the total displacement vector.

1. Horizontal Components:

   • $0 + 2 + (-2.12) = -0.12 \, \text{miles}$ (net movement slightly to the West).

2. Vertical Components:

   • $1 + 0 + 2.12 = 3.12 \, \text{miles}$ (net movement North).

Resultant Displacement:

• The total displacement vector has components: $(-0.12 \, \text{miles}, 3.12 \, \text{miles})$
• To find the magnitude (the length of the resultant vector), use the Pythagorean theorem: $\text{Resultant Magnitude} = \sqrt{(-0.12)^2 + (3.12)^2} = \sqrt{0.0144 + 9.7344} = \sqrt{9.7488} \approx 3.12 \, \text{miles}$
• To find the angle $\theta$ of the resultant vector relative to the horizontal (east-west line), use the tangent function: $\theta = \tan^{-1} \left( \frac{3.12}{0.12} \right) \approx \tan^{-1}(26) \approx 87.76^\circ$ This angle is nearly vertical, pointing mostly North.

Scale Diagram:

You can now draw the scale diagram by representing each movement vector with its respective magnitude and direction:

1. Draw a 1-mile vertical line (North).
2. Draw a 2-mile horizontal line to the right (East).
3. Draw a 3-mile line at a 45° angle going Northwest.
4. The resultant displacement is a vector connecting the start point to the end point.

Would you like me to assist with drawing the scale diagram or provide further clarification on the steps?

Related Questions:

1. How do you break any vector into components along the x and y axes?
2. How would the result change if the angle for the northwest movement was different, say 60°?
3. How do we calculate the magnitude of displacement when given its components?
4. What other methods could be used to solve for displacement apart from components?
5. How does the resultant displacement differ from total distance traveled?

Tip: When dealing with angles in vector problems, always pay attention to the signs (positive or negative) of the components based on the direction.