1. Given vectors

$\vec A = 2\hat i + 3\hat j - \hat k$ $\vec B = \hat i + 2\hat j + 2\hat k$

We need the magnitude of the component of $\vec A$ along $\vec B$.

2. Formula for scalar component (projection) of $\vec A$ along $\vec B$

$\text{Component of } \vec A \text{ along } \vec B = \frac{\vec A \cdot \vec B}{|\vec B|}$

Since the question asks for the magnitude, we take the absolute value if needed.

3. Find the dot product

$\vec A \cdot \vec B = (2)(1) + (3)(2) + (-1)(2)$

$= 2 + 6 - 2 = 6$

4. Find the magnitude of $\vec B$

$|\vec B| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1+4+4} = \sqrt{9} = 3$

5. Compute the component

$\frac{\vec A \cdot \vec B}{|\vec B|} = \frac{6}{3} = 2$

So, the magnitude of the component of $\vec A$ along $\vec B$ is

$\boxed{2\ \text{m}}$

6. Comparison with stored answer

Stored correct answer: $2$

Our derived answer is also $2$, so it agrees.