Example 3. A rigid body is spinning with an angular velocity of 3 radians per second about an axes parallel in 2i - 3k. Find the velocity of the particle at the point (3, -4, 1).

The magnitude of the angular velocity is given as $|\vec{\omega}| = 3$ rad/s.
The axis of rotation is parallel to the vector $\vec{u} = 2\hat{i} - 3\hat{k}$.
First, find the magnitude of $\vec{u}$:

$$|\vec{u}| = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}$$

The unit vector in the direction of the axis is $\hat{u} = \frac{\vec{u}}{|\vec{u}|} = \frac{2\hat{i} - 3\hat{k}}{\sqrt{13}}$.
Now, form the angular velocity vector:

$$\vec{\omega} = |\vec{\omega}| \hat{u} = 3 \left( \frac{2\hat{i} - 3\hat{k}}{\sqrt{13}} \right) = \frac{6}{\sqrt{13}}\hat{i} - \frac{9}{\sqrt{13}}\hat{k}$$

The axis passes through point $A = (1, -3, -1)$.
The particle is at point $P = (3, -4, 1)$.
The position vector $\vec{r}$ from a point on the axis (A) to the particle (P) is $\vec{r} = \vec{P} - \vec{A}$.

$$\vec{r} = (3-1)\hat{i} + (-4 - (-3))\hat{j} + (1 - (-1))\hat{k}$$ $$\vec{r} = 2\hat{i} + (-4+3)\hat{j} + (1+1)\hat{k}$$ $$\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k}$$

$$\vec{v} = \left( \frac{6}{\sqrt{13}}\hat{i} - \frac{9}{\sqrt{13}}\hat{k} \right) \times (2\hat{i} - \hat{j} + 2\hat{k})$$

Factor out $\frac{1}{\sqrt{13}}$:

$$\vec{v} = \frac{1}{\sqrt{13}} (6\hat{i} + 0\hat{j} - 9\hat{k}) \times (2\hat{i} - \hat{j} + 2\hat{k})$$

Compute the cross product:

$$(6\hat{i} - 9\hat{k}) \times (2\hat{i} - \hat{j} + 2\hat{k}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 0 & -9 \\ 2 & -1 & 2 \end{vmatrix}$$ $$= ((0)(2) - (-9)(-1))\hat{i} - ((6)(2) - (-9)(2))\hat{j} + ((6)(-1) - (0)(2))\hat{k}$$ $$= (0 - 9)\hat{i} - (12 - (-18))\hat{j} + (-6 - 0)\hat{k}$$ $$= -9\hat{i} - (12 + 18)\hat{j} - 6\hat{k}$$ $$= -9\hat{i} - 30\hat{j} - 6\hat{k}$$

Now, multiply by the scalar factor:

$$\vec{v} = \frac{1}{\sqrt{13}} (-9\hat{i} - 30\hat{j} - 6\hat{k})$$ $$\vec{v} = -\frac{9}{\sqrt{13}}\hat{i} - \frac{30}{\sqrt{13}}\hat{j} - \frac{6}{\sqrt{13}}\hat{k} \text{ m/s}$$

The velocity of the particle is $\boxed{-\frac{9}{\sqrt{13}}\hat{i} - \frac{30}{\sqrt{13}}\hat{j} - \frac{6}{\sqrt{13}}\hat{k} \text{ m/s}}$.