1. Sam travels 12.0 m 20.00 west of north. He then travels 20.0m [N 40.00E]. W
a) Draw a labeled vector diagram. Label all vectors with direction and magnitude.
(C: 4 marks)
b) Calculate Sam’s total displacement using the component method. Show your
work. (A: 8 marks)
c) Write a statement that answers the question. (C: 2 marks)

Let's break down the problem step by step. I'll help guide through the process of solving it.

### Part (a): Draw a labeled vector diagram

Sam's journey consists of two vectors:
1. **Vector 1:** 12.0 m, 20.0° west of north.
2. **Vector 2:** 20.0 m, 40.0° east of north.

To draw this diagram:
- Start by drawing **Vector 1**, which is 12.0 m long. Its direction is 20.0° west of north, so the vector will be slightly tilted to the left from the north (towards the west).
- Then, draw **Vector 2**, which is 20.0 m long. Its direction is 40.0° east of north, meaning it is slightly tilted to the right from the north (towards the east).

### Part (b): Calculate Sam’s total displacement using the component method

We'll break both vectors into their **x (east-west)** and **y (north-south)** components and then sum the components to get the resultant displacement.

#### Step 1: Break down each vector into components
For each vector, we will use trigonometry (sine and cosine functions):

\[
\text{For a vector with magnitude \( r \) at an angle \( \theta \):}
\]
- \( x \text{-component} = r \cdot \sin(\theta) \)
- \( y \text{-component} = r \cdot \cos(\theta) \)

#### **Vector 1** (12.0 m, 20.0° west of north):
- Angle is 20.0° west of north, so:
    - \( x_1 = -12.0 \cdot \sin(20.0^\circ) \)
    - \( y_1 = 12.0 \cdot \cos(20.0^\circ) \)
    
(Note: The x-component is negative because it is westward.)

#### **Vector 2** (20.0 m, 40.0° east of north):
- Angle is 40.0° east of north, so:
    - \( x_2 = 20.0 \cdot \sin(40.0^\circ) \)
    - \( y_2 = 20.0 \cdot \cos(40.0^\circ) \)

#### Step 2: Calculate components

Let's calculate these values using the trigonometric functions:

For Vector 1:
\[
x_1 = -12.0 \cdot \sin(20.0^\circ) = -4.10 \, \text{m} \quad \text{(westward)}
\]
\[
y_1 = 12.0 \cdot \cos(20.0^\circ) = 11.28 \, \text{m} \quad \text{(northward)}
\]

For Vector 2:
\[
x_2 = 20.0 \cdot \sin(40.0^\circ) = 12.86 \, \text{m} \quad \text{(eastward)}
\]
\[
y_2 = 20.0 \cdot \cos(40.0^\circ) = 15.32 \, \text{m} \quad \text{(northward)}
\]

#### Step 3: Sum the components

- Total **x-component**: \( x_{\text{total}} = x_1 + x_2 = -4.10 + 12.86 = 8.76 \, \text{m} \)
- Total **y-component**: \( y_{\text{total}} = y_1 + y_2 = 11.28 + 15.32 = 26.60 \, \text{m} \)

#### Step 4: Calculate the resultant displacement

The magnitude of the resultant displacement \( R \) can be found using the Pythagorean theorem:
\[
R = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2} = \sqrt{(8.76)^2 + (26.60)^2} = \sqrt{76.77 + 707.56} = \sqrt{784.33} = 28.00 \, \text{m}
\]

#### Step 5: Find the direction of the resultant displacement

The direction \( \theta \) can be found using the inverse tangent:
\[
\theta = \tan^{-1} \left( \frac{y_{\text{total}}}{x_{\text{total}}} \right) = \tan^{-1} \left( \frac{26.60}{8.76} \right) = \tan^{-1}(3.04) = 71.96^\circ
\]
So the angle is approximately **72.0° north of east**.

### Part (c): Write a statement that answers the question

Sam's total displacement is **28.0 m** at an angle of **72.0° north of east**.

---

### Do you want more details or have any questions? Here are 5 related questions to help expand your understanding:

1. How can you apply the component method for displacement in 3D space?
2. What is the significance of vector addition in real-world physics problems?
3. How would the displacement calculation change if there was a third vector?
4. What role do trigonometric functions play in breaking down vectors?
5. How can you convert between polar and rectangular coordinates for vectors?

#### Tip: When adding vectors, always break them into components and pay attention to the direction!

Sam travels 12.0 m 20.0° west of north. He then travels 20.0 m [N 40.0° E]. Calculate Sam's total displacement using the component method and draw a labeled vector diagram.

Solve a physics problem where Sam travels along two vectors: 12.0 m 20.0° west of north and 20.0 m [N 40.0° E]. Learn how to calculate total displacement using the component method with step-by-step instructions. Suitable for Grades 10-12.

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