Step-by-step Solution:

Correct Answer: A ($3\hat{i} - 3\hat{j} + 2\hat{k}$)

The solution involves two main steps: first, finding the direction of the required vector, and second, scaling it to the correct magnitude.

Step 1: Find the Direction Vector

A vector that is parallel to $2\vec{a} - \vec{b} + 3\vec{c}$ will have the same direction. Let's first calculate this resultant vector, which we'll call $\vec{v}_{dir}$.

Given vectors:

• $\vec{a} = \hat{i} + \hat{j} + \hat{k}$
• $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$
• $\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$

$\vec{v}_{dir} = 2\vec{a} - \vec{b} + 3\vec{c}$

$\vec{v}_{dir} = 2(\hat{i} + \hat{j} + \hat{k}) - (2\hat{i} - \hat{j} + 3\hat{k}) + 3(\hat{i} - 2\hat{j} + \hat{k})$

$\vec{v}_{dir} = (2\hat{i} + 2\hat{j} + 2\hat{k}) - (2\hat{i} - \hat{j} + 3\hat{k}) + (3\hat{i} - 6\hat{j} + 3\hat{k})$

Now, combine the components:

$\vec{v}_{dir} = (2 - 2 + 3)\hat{i} + (2 - (-1) - 6)\hat{j} + (2 - 3 + 3)\hat{k}$

$\vec{v}_{dir} = 3\hat{i} - 3\hat{j} + 2\hat{k}$

This is the direction of the vector we are looking for.

Step 2: Find the Vector with the Required Magnitude

We need a vector with a magnitude of $\sqrt{22}$. Let's first find the magnitude of our direction vector, $|\vec{v}_{dir}|$.

$|\vec{v}_{dir}| = \sqrt{(3)^2 + (-3)^2 + (2)^2}$

$|\vec{v}_{dir}| = \sqrt{9 + 9 + 4} = \sqrt{22}$

The magnitude of our direction vector is already $\sqrt{22}$, which is the required magnitude. Therefore, the vector we are looking for is exactly $\vec{v}_{dir}$.

Required Vector = $3\hat{i} - 3\hat{j} + 2\hat{k}$