A student observed an ant crawling on a wall. He determined the ant's displacement by using a protractor and ruler. The displacements are as follows 2.5 cm east, 3 cm, 35° north of east and 4.1 cm, 18º west of north

To find the resultant displacement of the ant, we can break down each displacement into its components along the east (x-axis) and north (y-axis) directions, sum these components, and then calculate the magnitude and direction of the resultant vector.

### Step 1: Breakdown the Displacements into Components
1. **First displacement: 2.5 cm east**
   - \( x_1 = 2.5 \, \text{cm} \)
   - \( y_1 = 0 \, \text{cm} \)

2. **Second displacement: 3 cm, 35° north of east**
   - \( x_2 = 3 \times \cos(35^\circ) \)
   - \( y_2 = 3 \times \sin(35^\circ) \)

3. **Third displacement: 4.1 cm, 18° west of north**
   - \( x_3 = -4.1 \times \sin(18^\circ) \)
   - \( y_3 = 4.1 \times \cos(18^\circ) \)

### Step 2: Calculate the Components
Let’s compute these values.

\[
x_2 = 3 \times \cos(35^\circ) \approx 3 \times 0.8192 = 2.4576 \, \text{cm}
\]
\[
y_2 = 3 \times \sin(35^\circ) \approx 3 \times 0.5736 = 1.7208 \, \text{cm}
\]
\[
x_3 = -4.1 \times \sin(18^\circ) \approx -4.1 \times 0.3090 = -1.2669 \, \text{cm}
\]
\[
y_3 = 4.1 \times \cos(18^\circ) \approx 4.1 \times 0.9511 = 3.8995 \, \text{cm}
\]

### Step 3: Sum the Components
Now, sum the x and y components:

\[
\text{Total } x = x_1 + x_2 + x_3 = 2.5 + 2.4576 - 1.2669 = 3.6907 \, \text{cm}
\]
\[
\text{Total } y = y_1 + y_2 + y_3 = 0 + 1.7208 + 3.8995 = 5.6203 \, \text{cm}
\]

### Step 4: Calculate the Magnitude and Direction of the Resultant Displacement
The magnitude \( R \) of the resultant displacement is given by:
\[
R = \sqrt{(\text{Total } x)^2 + (\text{Total } y)^2}
\]
\[
R = \sqrt{(3.6907)^2 + (5.6203)^2} \approx \sqrt{13.6264 + 31.5861} = \sqrt{45.2125} \approx 6.724 \, \text{cm}
\]

The direction \( \theta \) relative to the east (x-axis) is:
\[
\theta = \tan^{-1} \left(\frac{\text{Total } y}{\text{Total } x}\right) = \tan^{-1} \left(\frac{5.6203}{3.6907}\right) \approx \tan^{-1} (1.5223) \approx 56.41^\circ \text{ north of east}
\]

### Final Result:
The resultant displacement of the ant is approximately **6.72 cm** at an angle of **56.41° north of east**.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How would the resultant displacement change if one of the displacements was in the opposite direction?
2. What would happen if the third displacement was doubled?
3. How can we use vectors to represent forces instead of displacements?
4. What is the significance of vector components in physics?
5. How would the calculation differ if the ant moved in a three-dimensional space?
6. Can you calculate the resultant displacement if the directions were given in different units, such as radians?
7. How does the Pythagorean theorem relate to the magnitude of vectors?
8. What are some real-life applications of vector addition?

**Tip:** When dealing with vector addition, always break the vectors into components and treat each component along the axis separately. This simplifies the calculation process.

Learn how to calculate the resultant displacement of an ant crawling on a wall using vector addition. This example covers multiple displacements with specific angles and distances, applying formulas for vector components, magnitude, and direction. Suitable for high school students studying vector addition.

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