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The problem states that a delivery truck drives 1 mile north, 2 miles east, and 3 miles northwest, and asks to determine the resultant displacement.

Let’s solve this step by step, focusing on part **(b) by using components**:

### Step 1: Breaking the motion into components
- **1 mile north**: This movement is purely in the y-direction (positive y). So, the displacement vector is:
  \[
  \vec{A} = (0, 1) \text{ miles}
  \]
- **2 miles east**: This movement is purely in the x-direction (positive x). So, the displacement vector is:
  \[
  \vec{B} = (2, 0) \text{ miles}
  \]
- **3 miles northwest**: Northwest means the direction is 45° between west and north. We decompose it into x and y components:
  - x-component: \( 3 \times \cos(135^\circ) = 3 \times (-\frac{1}{\sqrt{2}}) = -\frac{3}{\sqrt{2}} \)
  - y-component: \( 3 \times \sin(135^\circ) = 3 \times \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} \)
  So, the displacement vector is:
  \[
  \vec{C} = \left(-\frac{3}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right)
  \]

### Step 2: Adding the components
Now, we sum the x and y components of the three vectors:

- **x-component total**:
  \[
  0 + 2 + \left(-\frac{3}{\sqrt{2}}\right) = 2 - \frac{3}{\sqrt{2}} \approx 2 - 2.12 = -0.12
  \]
- **y-component total**:
  \[
  1 + 0 + \frac{3}{\sqrt{2}} = 1 + 2.12 = 3.12
  \]

### Step 3: Resultant displacement
The resultant displacement vector \( \vec{R} \) is:
\[
\vec{R} = (-0.12, 3.12)
\]
The magnitude of this resultant vector is:
\[
|\vec{R}| = \sqrt{(-0.12)^2 + (3.12)^2} \approx \sqrt{0.0144 + 9.7344} = \sqrt{9.7488} \approx 3.12 \text{ miles}
\]

### Step 4: Direction of displacement
The direction (angle) relative to the x-axis can be found using:
\[
\theta = \tan^{-1}\left(\frac{3.12}{-0.12}\right) \approx \tan^{-1}(-26) \approx 92.64^\circ
\]
This indicates the direction is almost due north.

---

Would you like me to explain part (a) on how to draw the scale diagram? Let me know if you need further clarification!

Here are 5 related questions for practice:
1. How do you calculate the resultant of two perpendicular vectors?
2. What is the significance of the angle between vectors when decomposing them into components?
3. How would the displacement change if the truck moved southwest instead of northwest?
4. Why is the Pythagorean theorem useful in vector calculations?
5. Can you calculate the angle of resultant displacement using a different method?

**Tip**: When breaking vectors into components, always pay close attention to the signs of the trigonometric functions in different quadrants.

A delivery truck drives 1 mile north, 2 miles east, and 3 miles northwest. Determine the resultant displacement by drawing a scale diagram and using components.

Learn how to calculate the resultant displacement of a delivery truck traveling 1 mile north, 2 miles east, and 3 miles northwest. This detailed solution includes vector addition, trigonometric decomposition, and the Pythagorean theorem. Suitable for high school students.

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