To determine the truck's resultant displacement, we can break the motion into vector components and then combine them. Here's how to approach this problem step-by-step:

1. Break Down the Motion

The truck moves in four segments:

1. 3 miles north – A vertical vector pointing upward.
2. 4 miles east – A horizontal vector pointing to the right.
3. 3 miles south – A vertical vector pointing downward.
4. 2 miles northeast – A diagonal vector at a 45-degree angle.

2. Draw the Scale Diagram

To represent the motion graphically:

• Start at the origin (0, 0).

• Draw each vector based on the direction and magnitude using a consistent scale (e.g., 1 mile = 1 unit).

• First, draw a vector 3 units upward for the northward motion.

• Next, draw a vector 4 units to the right for the eastward motion.

• Then, draw a vector 3 units downward for the southward motion.

• Finally, draw a vector of 2 units at a 45-degree angle northeast (which involves breaking it into x and y components).

3. Use Components to Find Resultant

We'll break each vector into x (east-west) and y (north-south) components:

• 3 miles north:

  • x-component = 0 (no east-west movement)
  • y-component = +3 miles (north)

• 4 miles east:

  • x-component = +4 miles (east)
  • y-component = 0 (no north-south movement)

• 3 miles south:

  • x-component = 0 (no east-west movement)
  • y-component = -3 miles (south)

• 2 miles northeast:

  • Since the vector is at a 45° angle, the components are:
    • x-component = $2 \cos 45^\circ = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2} \approx 1.414$ miles
    • y-component = $2 \sin 45^\circ = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2} \approx 1.414$ miles

4. Add the Components

• Total x-component = $0 + 4 + 0 + 1.414 = 5.414$ miles
• Total y-component = $3 + 0 - 3 + 1.414 = 1.414$ miles

5. Calculate the Resultant Displacement

Now, we calculate the magnitude of the resultant vector using the Pythagorean theorem:

$\text{Resultant Displacement} = \sqrt{(5.414)^2 + (1.414)^2}$

$= \sqrt{29.305 + 1.999}$

$= \sqrt{31.304} \approx 5.6 \, \text{miles}$

The direction (angle θ) of the resultant vector can be found using:

$\tan \theta = \frac{\text{Total y-component}}{\text{Total x-component}} = \frac{1.414}{5.414}$

$\theta = \tan^{-1}(0.261) \approx 14.7^\circ \, \text{north of east}$

Result:

The truck's resultant displacement is approximately 5.6 miles at an angle of 14.7° north of east.

Would you like to see more details or a visual representation of the diagram?

Here are 5 related questions to further explore:

1. What if the truck traveled 5 miles west instead of east?
2. How would the displacement change if the northeast angle was different?
3. What are the components of a vector that goes northwest at 3 miles?
4. How do you calculate resultant displacement for more than four vectors?
5. What is the difference between distance traveled and displacement?

Tip: Always resolve diagonal vectors into their components for accurate results in vector addition.