Given the vectors $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$, and $\vec{c} = -\hat{i} + 3\hat{j} + 2\hat{k}$, we are asked to find $(\vec{a} + \vec{b}) \cdot (\vec{b} - \vec{c})$.

Step 1: Calculate $\vec{a} + \vec{b}$

$\vec{a} + \vec{b} = (2\hat{i} - 2\hat{j} + \hat{k}) + (\hat{j} - \hat{k}) = 2\hat{i} + (-2 + 1)\hat{j} + (1 - 1)\hat{k} = 2\hat{i} - \hat{j}$

Step 2: Calculate $\vec{b} - \vec{c}$

$\vec{b} - \vec{c} = (\hat{j} - \hat{k}) - (-\hat{i} + 3\hat{j} + 2\hat{k}) = 0\hat{i} + (1 - 3)\hat{j} + (-1 - 2)\hat{k} = \hat{i} - 2\hat{j} - 3\hat{k}$

Step 3: Calculate the dot product $(\vec{a} + \vec{b}) \cdot (\vec{b} - \vec{c})$

$(2\hat{i} - \hat{j}) \cdot (\hat{i} - 2\hat{j} - 3\hat{k}) = (2 \times 1) + (-1 \times -2) + (0 \times -3) = 2 + 2 + 0 = 4$

Thus, the result of $(\vec{a} + \vec{b}) \cdot (\vec{b} - \vec{c})$ is $\boxed{4}$.

Would you like any further details or explanations?

Here are some related questions:

1. How do you find the cross product of two vectors?
2. What are the properties of dot products?
3. How do vector addition and subtraction work geometrically?
4. How can we use vectors in physics problems?
5. What is the significance of orthogonal vectors?

Tip: Remember that the dot product of two vectors results in a scalar quantity, which can provide insights about the angle between the vectors and their magnitudes.