a truck drives 3 miles north then 4 miles east then 3 miles south then 2 miles northeast, determine the resultant dis placement by drawing a scale diagram by using components

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.

A truck drives 3 miles north then 4 miles east then 3 miles south then 2 miles northeast, determine the resultant displacement by drawing a scale diagram by using components.

Learn how to calculate the resultant displacement of a truck that drives 3 miles north, 4 miles east, 3 miles south, and 2 miles northeast. This problem uses vector addition, the Pythagorean theorem, and trigonometry to find the displacement, suitable for Grades 9-11.

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