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The question asks us to determine \( A + B^2 \), where:

- \( \vec{u} = (6,4) \)
- \( \vec{v} = (-3,2) \)
- \( A = \vec{u} \cdot \vec{v} \) (dot product of \( \vec{u} \) and \( \vec{v} \))
- \( B = \|\vec{v}\| \) (magnitude of \( \vec{v} \))

We will proceed step by step:

### Step 1: Calculate the dot product \( A = \vec{u} \cdot \vec{v} \)
\[
A = (6)(-3) + (4)(2) = -18 + 8 = -10
\]

### Step 2: Calculate the magnitude \( B = \|\vec{v}\| \)
\[
B = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}
\]

### Step 3: Calculate \( B^2 \)
\[
B^2 = (\sqrt{13})^2 = 13
\]

### Step 4: Determine \( A + B^2 \)
\[
A + B^2 = -10 + 13 = 3
\]

Thus, none of the provided options seem to match \( A + B^2 = 3 \). There might be a typo or mismatch in the question options.

Do you want to clarify further details? 

Here are 5 related questions:
1. How is the dot product used in other vector operations?
2. How do we calculate the magnitude of a vector in higher dimensions?
3. What are the geometric interpretations of the dot product and vector magnitude?
4. Can \( A + B^2 \) represent a physical quantity in some context?
5. How does the result change if we use different vectors?

Tip: Double-check calculations when the given options don't match the result, as there could be small errors in either the problem or the choices provided.

Dado el vector \vec{u} = (6, 4) y el vector \vec{v} = (-3, 2); siendo A = \vec{u} \cdot \vec{v} y B = ||\vec{v}||, determine A + B^2.

Learn how to calculate the dot product of vectors and the magnitude of a vector. Given \vec{u} = (6,4) and \vec{v} = (-3,2), this step-by-step solution helps determine A + B^2, where A is the dot product of the vectors and B is the magnitude of \vec{v}.

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