Given that \(|\vec{a}| = 1\), \(|\vec{b}| = \sqrt{3}\), and \(\vec{a}\) and \(\vec{b}\) are perpendicular, we need to evaluate:

(a) \(|(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b})|\)  
(b) \(|(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b})|\)

### Solution:

#### Part (a):
First, expand the dot product:
\[
(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b}
\]

Since \(\vec{a} \cdot \vec{b} = 0\) (they are perpendicular):
\[
(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b}
\]

Using \(|\vec{a}|^2 = \vec{a} \cdot \vec{a} = 1\) and \(|\vec{b}|^2 = \vec{b} \cdot \vec{b} = 3\):
\[
(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 1 - 3 = -2
\]

So, the magnitude is:
\[
|(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b})| = |-2| = 2
\]

#### Part (b):
Expand the dot product:
\[
(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b}) = 2\vec{a} \cdot \vec{a} - 4\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - 2\vec{b} \cdot \vec{b}
\]

Again, using \(\vec{a} \cdot \vec{b} = 0\):
\[
(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b}) = 2\vec{a} \cdot \vec{a} - 2\vec{b} \cdot \vec{b}
\]

Substitute the magnitudes:
\[
(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b}) = 2(1) - 2(3) = 2 - 6 = -4
\]

So, the magnitude is:
\[
|(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b})| = |-4| = 4
\]

### Final Answers:
- (a) \(|(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b})| = 2\)
- (b) \(|(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b})| = 4\)

Would you like any additional details or explanations?

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Here are some related questions to consider:

1. What are the properties of dot products for perpendicular vectors?
2. How does the magnitude of vectors affect the dot product?
3. Can dot products be used to determine the angle between two vectors?
4. What happens to the dot product if one of the vectors is multiplied by a scalar?
5. How can we extend this problem to three dimensions?
6. What is the geometric interpretation of the result in part (a)?
7. How would the answers change if \(\vec{a}\) and \(\vec{b}\) were not perpendicular?
8. How does vector addition affect the overall magnitude of the resulting vector?

**Tip:** Always check the angle between vectors when dealing with dot products, as this can simplify the calculation significantly.

Explore how to evaluate dot products involving vectors with specific magnitudes and perpendicularity conditions. Understand the properties of dot products for perpendicular vectors and their application in vector algebra. Suitable for advanced high school students and beyond.

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