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To find the resultant vector of the two given forces, we'll break each vector into its horizontal (x) and vertical (y) components and then add these components.

### Step 1: Resolve each force into components

1. **First vector**: \( 3 \, \text{N} \) at \( 73^\circ \) from the horizontal.
   - **Horizontal component**: \( 3 \cos(73^\circ) \)
   - **Vertical component**: \( 3 \sin(73^\circ) \)

2. **Second vector**: \( 1 \, \text{N} \) at \( 90^\circ \) from the horizontal.
   - **Horizontal component**: \( 1 \cos(90^\circ) = 0 \)
   - **Vertical component**: \( 1 \sin(90^\circ) = 1 \, \text{N} \)

Using approximate values for trigonometric functions:
- \( \cos(73^\circ) \approx 0.2924 \)
- \( \sin(73^\circ) \approx 0.9563 \)

#### Components of the vectors:
1. For \( 3 \, \text{N} \):
   - \( x_1 = 3 \times 0.2924 \approx 0.8772 \, \text{N} \)
   - \( y_1 = 3 \times 0.9563 \approx 2.8689 \, \text{N} \)

2. For \( 1 \, \text{N} \):
   - \( x_2 = 0 \, \text{N} \)
   - \( y_2 = 1 \, \text{N} \)

### Step 2: Add the components

- **Total horizontal component** \( (x_{\text{total}}) = x_1 + x_2 = 0.8772 + 0 = 0.8772 \, \text{N} \)
- **Total vertical component** \( (y_{\text{total}}) = y_1 + y_2 = 2.8689 + 1 = 3.8689 \, \text{N} \)

### Step 3: Calculate the magnitude and direction of the resultant vector

1. **Magnitude**:
   \[
   R = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2}
   = \sqrt{(0.8772)^2 + (3.8689)^2}
   \approx \sqrt{0.7695 + 14.9666} \approx \sqrt{15.7361} \approx 3.97 \, \text{N}
   \]

2. **Direction** \( \theta \) (angle with respect to the horizontal):
   \[
   \theta = \tan^{-1} \left( \frac{y_{\text{total}}}{x_{\text{total}}} \right)
   = \tan^{-1} \left( \frac{3.8689}{0.8772} \right) \approx \tan^{-1}(4.41) \approx 77.5^\circ
   \]

### Result

The resultant vector has a magnitude of approximately **3.97 N** and makes an angle of **77.5°** with the horizontal.

Would you like further details on any part of this solution?

### Relative Questions
1. How do you calculate vector components using trigonometry?
2. Why do we use the tangent function to find the angle of the resultant vector?
3. How does changing the angle of a force affect the resultant vector?
4. What are some applications of vector addition in physics?
5. How would the result change if the forces were in opposite directions?

### Tip
When dealing with vectors, always draw a clear diagram to visualize the directions and angles involved—it makes component calculations easier.

Find the resultant vector of a 3 N force at 73° from the horizontal and a 1 N force at 90° from the horizontal.

Calculate the resultant vector of two forces: 3 N at 73° and 1 N at 90°. This physics problem involves vector addition, trigonometric calculations, and finding both magnitude and angle of the resultant vector. Suitable for high school students in Grades 10-12.

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