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The problem is asking for the magnitude and direction of the vector sum **\(\vec{v} + \vec{w}\)**. Let's break this down step by step.

### Information Given:
- **Vector \(\vec{v}\)**: 
  - Magnitude = 4
  - Direction = 310° (clockwise from the positive \(x\)-axis)
  
- **Vector \(\vec{w}\)**: 
  - Magnitude = 7
  - Direction = 50° (counterclockwise from the positive \(x\)-axis)

### Steps to Solve:

1. **Convert the vector components:**
   - The \(x\)-component of each vector is given by \(\text{magnitude} \times \cos(\text{angle})\).
   - The \(y\)-component of each vector is given by \(\text{magnitude} \times \sin(\text{angle})\).
   
   Let's calculate the components for both vectors:

   - For **\(\vec{v}\)** (magnitude 4, angle 310°):
     \[
     v_x = 4 \times \cos(310^\circ)
     \]
     \[
     v_y = 4 \times \sin(310^\circ)
     \]
   - For **\(\vec{w}\)** (magnitude 7, angle 50°):
     \[
     w_x = 7 \times \cos(50^\circ)
     \]
     \[
     w_y = 7 \times \sin(50^\circ)
     \]

2. **Sum the components:**
   - Add the \(x\)-components and \(y\)-components of \(\vec{v}\) and \(\vec{w}\):
     \[
     x_{\text{total}} = v_x + w_x
     \]
     \[
     y_{\text{total}} = v_y + w_y
     \]

3. **Find the magnitude of the resultant vector \(\vec{v} + \vec{w}\):**
   \[
   |\vec{v} + \vec{w}| = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2}
   \]

4. **Find the direction \(\theta\) of the resultant vector:**
   \[
   \theta = \tan^{-1}\left(\frac{y_{\text{total}}}{x_{\text{total}}}\right)
   \]

Let me calculate these values for you.The magnitude of the resultant vector \(\vec{v} + \vec{w}\) is approximately **7.43**.

The direction of the resultant vector is approximately **18.01°** counterclockwise from the positive \(x\)-axis.

Would you like further details or clarification on any steps? Here are a few related questions to deepen understanding:

1. How do the \(x\)- and \(y\)-components affect the magnitude of a vector?
2. Why do we need to convert degrees to radians in trigonometric calculations?
3. What is the significance of using \(\tan^{-1}\) to find the angle of a vector?
4. How would the result change if one of the vectors were negative?
5. What is the geometric interpretation of adding two vectors?

**Tip:** When dealing with vector addition, always keep the angle direction in mind, as it affects the signs of the components.

Urči velikost a směrový úhel θ vektoru \(\vec{v} + \vec{w}\).

Solve the vector addition problem by calculating the magnitude and direction of \(\vec{v} + \vec{w}\). Learn about vector components, trigonometric calculations, and how to apply the Pythagorean theorem and inverse tangent function. Suitable for high school students.

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