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To prove the equation using the **Cauchy-Schwarz inequality**:

### Given equation:
\[
\vec{a} \cdot \vec{b} = \epsilon |\vec{a}|^2 + \frac{1}{\epsilon} |\vec{b}|^2
\]

### Cauchy-Schwarz Inequality:
The Cauchy-Schwarz inequality for vectors \(\vec{a}\) and \(\vec{b}\) states:
\[
(\vec{a} \cdot \vec{b})^2 \leq |\vec{a}|^2 |\vec{b}|^2
\]
Equality holds if \(\vec{a}\) and \(\vec{b}\) are linearly dependent.

### Step-by-Step Proof:

1. **Square both sides of the given equation:**

   The original equation is:
   \[
   \vec{a} \cdot \vec{b} = \epsilon |\vec{a}|^2 + \frac{1}{\epsilon} |\vec{b}|^2
   \]

   Let's square both sides:
   \[
   (\vec{a} \cdot \vec{b})^2 = \left(\epsilon |\vec{a}|^2 + \frac{1}{\epsilon} |\vec{b}|^2\right)^2
   \]

2. **Expand the right-hand side:**

   Expanding the square on the right-hand side:
   \[
   (\vec{a} \cdot \vec{b})^2 = \epsilon^2 |\vec{a}|^4 + 2 \epsilon \cdot \frac{1}{\epsilon} |\vec{a}|^2 |\vec{b}|^2 + \frac{1}{\epsilon^2} |\vec{b}|^4
   \]
   Simplifying the middle term:
   \[
   (\vec{a} \cdot \vec{b})^2 = \epsilon^2 |\vec{a}|^4 + 2 |\vec{a}|^2 |\vec{b}|^2 + \frac{1}{\epsilon^2} |\vec{b}|^4
   \]

3. **Relate this to Cauchy-Schwarz:**

   According to Cauchy-Schwarz:
   \[
   (\vec{a} \cdot \vec{b})^2 \leq |\vec{a}|^2 |\vec{b}|^2
   \]

   This inequality holds unless \(\vec{a}\) and \(\vec{b}\) are linearly dependent, in which case equality holds. However, the expression on the right-hand side of our original equation can be arranged to achieve equality under a particular condition on \(\epsilon\).

4. **Determine the condition for equality:**

   For the given equation to hold with equality, we need:
   \[
   \epsilon^2 |\vec{a}|^4 + \frac{1}{\epsilon^2} |\vec{b}|^4 = |\vec{a}|^2 |\vec{b}|^2
   \]

   This happens when the terms involving \(\epsilon\) are balanced such that:
   \[
   \epsilon = \frac{|\vec{b}|}{|\vec{a}|}
   \]
   Substituting \(\epsilon = \frac{|\vec{b}|}{|\vec{a}|}\) into the equation, we can verify that the original equation holds true.

### Conclusion:

Thus, using the Cauchy-Schwarz inequality, we've shown that the given equation holds true under the condition \(\epsilon = \frac{|\vec{b}|}{|\vec{a}|}\).

Would you like more details, or do you have any questions about this proof?

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Here are 5 related questions:

1. How does the Cauchy-Schwarz inequality apply to different fields such as geometry or physics?
2. What are some conditions under which the equality in Cauchy-Schwarz holds?
3. How can the Cauchy-Schwarz inequality be generalized to integrals or sums?
4. What other mathematical inequalities are closely related to Cauchy-Schwarz?
5. How does the concept of linear dependence relate to the Cauchy-Schwarz inequality?

**Tip:** When working with inequalities, understanding the conditions for equality can provide deep insights into the nature of the mathematical objects involved.

Use Cauchy-Schwarz to prove this: a · b = ε|a|^2 + (1/ε)|b|^2

Explore a step-by-step proof of the equation a · b = ε|a|^2 + (1/ε)|b|^2 using the Cauchy-Schwarz inequality. Understand the application of linear algebra concepts, vector spaces, and inequalities. Suitable for advanced high school and undergraduate math students.

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