1. Define the Vectors

Angular Velocity \( \vec{\omega} \): The axis is in the x-y plane. It makes 45° with the +y axis and 45° with the -x axis. This corresponds to the direction bisecting the second quadrant.

\[ \vec{\omega} = \omega_0 \left( -\cos(45^\circ)\hat{i} + \sin(45^\circ)\hat{j} \right) = \frac{\omega_0}{\sqrt{2}} (-\hat{i} + \hat{j}) \]

Position Vector \( \vec{r} \): The point is on the positive z-axis at distance z.

\[ \vec{r} = z \hat{k} \]

2. Calculate Cross Product \[ \vec{v} = \vec{\omega} \times \vec{r} \] \[ \vec{v} = \left[ \frac{\omega_0}{\sqrt{2}} (-\hat{i} + \hat{j}) \right] \times (z \hat{k}) \] \[ \vec{v} = \frac{\omega_0 z}{\sqrt{2}} \left[ (-\hat{i} \times \hat{k}) + (\hat{j} \times \hat{k}) \right] \]

3. Apply Vector Rules

Using the standard unit vector cross products:

• \( \hat{i} \times \hat{k} = -\hat{j} \implies -\hat{i} \times \hat{k} = \hat{j} \)
• \( \hat{j} \times \hat{k} = \hat{i} \)

\[ \vec{v} = \frac{\omega_0 z}{\sqrt{2}} (\hat{j} + \hat{i}) \]