Step 1: Find the sum of the vectors $\vec{a}$ and $\vec{b}$

The given vectors are:

$\vec{a} = -4\hat{i} - 4\hat{j}, \quad \vec{b} = 7\hat{i} - 2\hat{j}.$

The sum of two vectors $\vec{a} + \vec{b}$ is calculated by adding their components:

$\vec{a} + \vec{b} = (-4 + 7)\hat{i} + (-4 - 2)\hat{j}.$

Simplify the components:

$\vec{a} + \vec{b} = 3\hat{i} - 6\hat{j}.$

Step 2: Find the magnitude $|\vec{a} + \vec{b}|$

The magnitude of a vector $\vec{v} = x\hat{i} + y\hat{j}$ is given by:

$|\vec{v}| = \sqrt{x^2 + y^2}.$

For $\vec{a} + \vec{b} = 3\hat{i} - 6\hat{j}$:

$|\vec{a} + \vec{b}| = \sqrt{3^2 + (-6)^2}.$

Simplify:

$|\vec{a} + \vec{b}| = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}.$

Final Results:

1. The sum of the vectors: $\vec{a} + \vec{b} = 3\hat{i} - 6\hat{j}.$

2. The magnitude of the resultant vector: $|\vec{a} + \vec{b}| = 3\sqrt{5}.$

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Would you like additional details or examples?
Here are 5 related questions to expand your understanding:

1. How do you subtract vectors and calculate the magnitude?
2. Can you represent the resultant vector in polar form?
3. What is the direction (angle) of the resultant vector?
4. How do you add vectors in three-dimensional space?
5. Can you explain how vector addition is applied in physics problems?

Tip: When dealing with vectors, always ensure to break them into components for easier calculations.