Given vectors a = 2i − 3j + k and b = -3i + j - 2k, find the magnitude of 2a - 3b.

Multiply each component of $\vec{a}$ by 2:

$$2\vec{a} = 2(2\vec{i} - 3\vec{j} + \vec{k})$$ $$2\vec{a} = 4\vec{i} - 6\vec{j} + 2\vec{k}$$

Multiply each component of $\vec{b}$ by 3:

$$3\vec{b} = 3(-3\vec{i} + \vec{j} - 2\vec{k})$$ $$3\vec{b} = -9\vec{i} + 3\vec{j} - 6\vec{k}$$

Subtract the components of $3\vec{b}$ from the corresponding components of $2\vec{a}$:

$$2\vec{a} - 3\vec{b} = (4\vec{i} - 6\vec{j} + 2\vec{k}) - (-9\vec{i} + 3\vec{j} - 6\vec{k})$$ $$2\vec{a} - 3\vec{b} = (4 - (-9))\vec{i} + (-6 - 3)\vec{j} + (2 - (-6))\vec{k}$$ $$2\vec{a} - 3\vec{b} = (4 + 9)\vec{i} + (-6 - 3)\vec{j} + (2 + 6)\vec{k}$$ $$2\vec{a} - 3\vec{b} = 13\vec{i} - 9\vec{j} + 8\vec{k}$$

For a vector $V = x\vec{i} + y\vec{j} + z\vec{k}$, its magnitude is given by $|V| = \sqrt{x^2 + y^2 + z^2}$.
Let $V = 2\vec{a} - 3\vec{b} = 13\vec{i} - 9\vec{j} + 8\vec{k}$.

$$|2\vec{a} - 3\vec{b}| = \sqrt{(13)^2 + (-9)^2 + (8)^2}$$ $$|2\vec{a} - 3\vec{b}| = \sqrt{169 + 81 + 64}$$ $$|2\vec{a} - 3\vec{b}| = \sqrt{314}$$

The magnitude of $2\vec{a} - 3\vec{b}$ is $\boxed{\sqrt{314}}$.