To find $3\mathbf{u} + 4\mathbf{v}$, we first calculate $3\mathbf{u}$ and $4\mathbf{v}$, and then add the resulting vectors.

Given: $\mathbf{u} = \langle 3, 5, -1 \rangle, \quad \mathbf{v} = \langle 2, -2, 3 \rangle$

1. Calculate $3\mathbf{u}$: $3\mathbf{u} = 3 \langle 3, 5, -1 \rangle = \langle 9, 15, -3 \rangle$

2. Calculate $4\mathbf{v}$: $4\mathbf{v} = 4 \langle 2, -2, 3 \rangle = \langle 8, -8, 12 \rangle$

3. Add $3\mathbf{u}$ and $4\mathbf{v}$: $3\mathbf{u} + 4\mathbf{v} = \langle 9 + 8, 15 + (-8), -3 + 12 \rangle = \langle 17, 7, 9 \rangle$

Thus, $3\mathbf{u} + 4\mathbf{v} = \langle 17, 7, 9 \rangle$.

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. What is a linear combination of vectors, and how does it relate to vector addition and scalar multiplication?
2. How can the concept of vector addition be used in physics to represent forces?
3. If you change the coefficients of $3\mathbf{u} + 4\mathbf{v}$, how would the resulting vector change?
4. What would be the result of subtracting $4\mathbf{v}$ from $3\mathbf{u}$?
5. How can you determine if two vectors are parallel using scalar multiplication?

Tip: When performing vector operations, always apply scalar multiplication first, followed by vector addition or subtraction.