To solve for $3\mathbf{u} - 6\mathbf{v}$, we will first scale each vector by the required constants and then subtract the results.

1. Given:

   • $\mathbf{u} = \langle 6, -2 \rangle$
   • $\mathbf{v} = \langle -4, 7 \rangle$

2. Calculate $3\mathbf{u}$: $3\mathbf{u} = 3 \cdot \langle 6, -2 \rangle = \langle 3 \cdot 6, 3 \cdot (-2) \rangle = \langle 18, -6 \rangle$

3. Calculate $6\mathbf{v}$: $6\mathbf{v} = 6 \cdot \langle -4, 7 \rangle = \langle 6 \cdot (-4), 6 \cdot 7 \rangle = \langle -24, 42 \rangle$

4. Now, find $3\mathbf{u} - 6\mathbf{v}$: $3\mathbf{u} - 6\mathbf{v} = \langle 18, -6 \rangle - \langle -24, 42 \rangle = \langle 18 - (-24), -6 - 42 \rangle$ $= \langle 18 + 24, -6 - 42 \rangle = \langle 42, -48 \rangle$

So, the result is: $3\mathbf{u} - 6\mathbf{v} = \langle 42, -48 \rangle$

Would you like more details or have any questions?

Here are some related questions for practice:

1. What is $2\mathbf{u} + 3\mathbf{v}$ given the same vectors $\mathbf{u}$ and $\mathbf{v}$?
2. How would you find the magnitude of $\mathbf{u}$ and $\mathbf{v}$?
3. How do you calculate the angle between two vectors?
4. What is $\mathbf{u} \cdot \mathbf{v}$ (the dot product of $\mathbf{u}$ and $\mathbf{v}$)?
5. How can you express $\mathbf{v}$ in terms of $\mathbf{u}$ if they are parallel?

Tip: Remember that when scaling a vector, each component of the vector gets multiplied by the scalar independently.