https://file.aiquickdraw.com/mathbot/file/173063673901328lsbt6f.jpg

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?

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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.

Calculate 3u - 6v where u = ⟨6, -2⟩ and v = ⟨-4, 7⟩.

Learn how to calculate the vector expression 3u - 6v where u = ⟨6, -2⟩ and v = ⟨-4, 7⟩. This solution involves scalar multiplication and vector subtraction, ideal for high school students studying vector operations.

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