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We are given the following information:
- \(\|\mathbf{u}\| = 4\)
- \(\|\mathbf{v}\| = 4\)
- \(\mathbf{u} \cdot \mathbf{v} = 3\)

We are asked to find \(\|\mathbf{u} + 3\mathbf{v}\|\).

### Step-by-step solution:

1. **Formula for the magnitude of the vector sum**:
   \[
   \|\mathbf{u} + 3\mathbf{v}\| = \sqrt{(\mathbf{u} + 3\mathbf{v}) \cdot (\mathbf{u} + 3\mathbf{v})}
   \]
   
   Expand the dot product:
   \[
   (\mathbf{u} + 3\mathbf{v}) \cdot (\mathbf{u} + 3\mathbf{v}) = \mathbf{u} \cdot \mathbf{u} + 2(3)(\mathbf{u} \cdot \mathbf{v}) + 9(\mathbf{v} \cdot \mathbf{v})
   \]

2. **Substitute known values**:
   - \(\mathbf{u} \cdot \mathbf{u} = \|\mathbf{u}\|^2 = 4^2 = 16\)
   - \(\mathbf{v} \cdot \mathbf{v} = \|\mathbf{v}\|^2 = 4^2 = 16\)
   - \(\mathbf{u} \cdot \mathbf{v} = 3\)

   Now substitute these into the expanded expression:
   \[
   \|\mathbf{u} + 3\mathbf{v}\|^2 = 16 + 2(3)(3) + 9(16)
   \]
   Simplifying:
   \[
   \|\mathbf{u} + 3\mathbf{v}\|^2 = 16 + 18 + 144 = 178
   \]

3. **Take the square root**:
   \[
   \|\mathbf{u} + 3\mathbf{v}\| = \sqrt{178} \approx 13.34
   \]

Thus, the magnitude of \(\|\mathbf{u} + 3\mathbf{v}\|\) is approximately **13.34**.

Would you like further details or have any other questions?

Here are 5 related questions:
1. How do we generalize the formula for the magnitude of the sum of two vectors?
2. Can you explain how dot products are used to calculate angles between vectors?
3. How does scaling a vector by a scalar affect its magnitude?
4. How would the calculation change if \(\mathbf{u}\) and \(\mathbf{v}\) were orthogonal?
5. What is the significance of the dot product being positive, negative, or zero?

**Tip:** When dealing with vector sums, always expand using the distributive property and remember that \(\mathbf{u} \cdot \mathbf{v}\) gives crucial information about the angle between the vectors.

If ||u|| = 4, ||v|| = 4 and u · v = 3, find ||u + 3v||, express your answer up to 2 decimal places.

Learn how to find the magnitude of the vector sum ||u + 3v|| when given the magnitudes ||u|| = 4, ||v|| = 4, and the dot product u · v = 3. This detailed step-by-step solution covers key vector concepts such as dot products and magnitude formulas. Suitable for students in Grades 11-12.

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