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.