Find 2𝘂, −3𝘃, 𝘂 + 𝘃, and 3𝘂 − 4𝘃 for the given vectors 𝘂 and 𝘃.

𝘂 = −6, 7,    𝘃 = 6, −9

2𝘂

 = 
 
−3𝘃

 = 
 
𝘂 + 𝘃

 = 3𝘂 − 4𝘃=

Given two vectors:

\[
\mathbf{u} = \begin{pmatrix} -6 \\ 7 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 6 \\ -9 \end{pmatrix}
\]

We are asked to find the following quantities:
- \( 2\mathbf{u} \)
- \( -3\mathbf{v} \)
- \( \mathbf{u} + \mathbf{v} \)
- \( 3\mathbf{u} - 4\mathbf{v} \)

### Step 1: Calculate \( 2\mathbf{u} \)
Multiply vector \( \mathbf{u} \) by 2:
\[
2\mathbf{u} = 2 \times \begin{pmatrix} -6 \\ 7 \end{pmatrix} = \begin{pmatrix} 2 \times -6 \\ 2 \times 7 \end{pmatrix} = \begin{pmatrix} -12 \\ 14 \end{pmatrix}
\]

### Step 2: Calculate \( -3\mathbf{v} \)
Multiply vector \( \mathbf{v} \) by -3:
\[
-3\mathbf{v} = -3 \times \begin{pmatrix} 6 \\ -9 \end{pmatrix} = \begin{pmatrix} -3 \times 6 \\ -3 \times -9 \end{pmatrix} = \begin{pmatrix} -18 \\ 27 \end{pmatrix}
\]

### Step 3: Calculate \( \mathbf{u} + \mathbf{v} \)
Add vectors \( \mathbf{u} \) and \( \mathbf{v} \):
\[
\mathbf{u} + \mathbf{v} = \begin{pmatrix} -6 \\ 7 \end{pmatrix} + \begin{pmatrix} 6 \\ -9 \end{pmatrix} = \begin{pmatrix} -6 + 6 \\ 7 + -9 \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix}
\]

### Step 4: Calculate \( 3\mathbf{u} - 4\mathbf{v} \)
Multiply vector \( \mathbf{u} \) by 3 and vector \( \mathbf{v} \) by 4, then subtract:
\[
3\mathbf{u} = 3 \times \begin{pmatrix} -6 \\ 7 \end{pmatrix} = \begin{pmatrix} -18 \\ 21 \end{pmatrix}
\]
\[
4\mathbf{v} = 4 \times \begin{pmatrix} 6 \\ -9 \end{pmatrix} = \begin{pmatrix} 24 \\ -36 \end{pmatrix}
\]
Now subtract:
\[
3\mathbf{u} - 4\mathbf{v} = \begin{pmatrix} -18 \\ 21 \end{pmatrix} - \begin{pmatrix} 24 \\ -36 \end{pmatrix} = \begin{pmatrix} -18 - 24 \\ 21 - (-36) \end{pmatrix} = \begin{pmatrix} -42 \\ 57 \end{pmatrix}
\]

### Final Results:
- \( 2\mathbf{u} = \begin{pmatrix} -12 \\ 14 \end{pmatrix} \)
- \( -3\mathbf{v} = \begin{pmatrix} -18 \\ 27 \end{pmatrix} \)
- \( \mathbf{u} + \mathbf{v} = \begin{pmatrix} 0 \\ -2 \end{pmatrix} \)
- \( 3\mathbf{u} - 4\mathbf{v} = \begin{pmatrix} -42 \\ 57 \end{pmatrix} \)

Do you have any questions or would you like more details?

### 5 Related Questions:
1. How do you multiply a scalar with a vector?
2. How is vector addition performed geometrically?
3. What is the difference between scalar multiplication and vector subtraction?
4. How do vectors behave under matrix transformations?
5. Can vectors of different dimensions be added?

### Tip:
When performing vector arithmetic, ensure that you are consistent with the dimensions of the vectors to avoid mistakes.

Find 2𝘂, −3𝘃, 𝘂 + 𝘃, and 3𝘂 − 4𝘃 for the given vectors 𝘂 and 𝘃.
𝘂 = −6, 7, 𝘃 = 6, −9.

Learn how to perform vector operations including scalar multiplication, vector addition, and subtraction with given vectors u = (-6, 7) and v = (6, -9). This problem covers key mathematical concepts in vector arithmetic, suitable for students in Grades 9-12.

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