what about if the same problem had a second part asking you to find the value of the magnitude of (a-2b)

To find the magnitude of the vector \(\mathbf{a} - 2\mathbf{b}\), we will use the formula for the magnitude of a vector in terms of its components. Given the vectors \(\mathbf{a}\) and \(\mathbf{b}\) and their magnitudes, let's proceed step-by-step.

### Step-by-Step Solution

1. **Vector Magnitude Formula:**
   The magnitude of a vector \(\mathbf{v}\) can be calculated using:
   \[
   |\mathbf{v}| = \sqrt{\mathbf{v} \cdot \mathbf{v}}
   \]

   Here, we want to find the magnitude of \(\mathbf{a} - 2\mathbf{b}\):
   \[
   |\mathbf{a} - 2\mathbf{b}| = \sqrt{(\mathbf{a} - 2\mathbf{b}) \cdot (\mathbf{a} - 2\mathbf{b})}
   \]

2. **Expand the Dot Product:**
   Let's compute \((\mathbf{a} - 2\mathbf{b}) \cdot (\mathbf{a} - 2\mathbf{b})\) using the distributive property:
   \[
   (\mathbf{a} - 2\mathbf{b}) \cdot (\mathbf{a} - 2\mathbf{b}) = \mathbf{a} \cdot \mathbf{a} - 2(\mathbf{a} \cdot \mathbf{b}) - 2(\mathbf{b} \cdot \mathbf{a}) + 4(\mathbf{b} \cdot \mathbf{b})
   \]

   Since \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\), this simplifies to:
   \[
   = \mathbf{a} \cdot \mathbf{a} - 4(\mathbf{a} \cdot \mathbf{b}) + 4(\mathbf{b} \cdot \mathbf{b})
   \]

3. **Substitute Known Values:**
   We know:
   - \(|\mathbf{a}|^2 = \mathbf{a} \cdot \mathbf{a} = 1^2 = 1\)
   - \(|\mathbf{b}|^2 = \mathbf{b} \cdot \mathbf{b} = 3^2 = 9\)
   - \(\mathbf{a} \cdot \mathbf{b} = \frac{3}{2}\)

   Substituting these into the equation:
   \[
   (\mathbf{a} - 2\mathbf{b}) \cdot (\mathbf{a} - 2\mathbf{b}) = 1 - 4\left(\frac{3}{2}\right) + 4(9)
   \]

   Simplify the expression:
   \[
   = 1 - 6 + 36
   \]

   \[
   = 1 + 30 = 31
   \]

4. **Calculate the Magnitude:**
   Now, the magnitude of \(\mathbf{a} - 2\mathbf{b}\) is:
   \[
   |\mathbf{a} - 2\mathbf{b}| = \sqrt{31}
   \]

### Final Answer

The exact value of the magnitude of \(\mathbf{a} - 2\mathbf{b}\) is \(\boxed{\sqrt{31}}\).

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### Additional Questions

1. How would the result change if the magnitudes of \(\mathbf{a}\) or \(\mathbf{b}\) were different?
2. What is the significance of the square root in the magnitude formula?
3. Can you find a vector \(\mathbf{c}\) such that \(\mathbf{c} = \mathbf{a} - 2\mathbf{b}\) in terms of components?
4. How do you interpret the result geometrically in three-dimensional space?
5. How would you compute the direction of the vector \(\mathbf{a} - 2\mathbf{b}\)?

**Tip:** When working with vector magnitudes and dot products, always ensure to substitute the correct values and carefully handle algebraic expressions to avoid mistakes.

Learn how to calculate the magnitude of vector a - 2b step-by-step with a detailed mathematical solution. This problem involves vectors, their magnitudes, and dot products, suitable for advanced high school students and beyond.

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