To find a unit vector that is perpendicular to two given vectors, let's denote these vectors as $\vec{A}$ and $\vec{B}$. A vector perpendicular to both $\vec{A}$ and $\vec{B}$ can be obtained using the cross product, $\vec{A} \times \vec{B}$.

Here are the steps to find the unit vector perpendicular to both vectors:

1. Find the cross product $\vec{A} \times \vec{B}$:
   If $\vec{A} = \langle a_1, a_2, a_3 \rangle$ and $\vec{B} = \langle b_1, b_2, b_3 \rangle$, then $\vec{A} \times \vec{B} = \langle a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \rangle$

2. Calculate the magnitude of $\vec{A} \times \vec{B}$:
   Let $\vec{C} = \vec{A} \times \vec{B} = \langle c_1, c_2, c_3 \rangle$. The magnitude $|\vec{C}|$ is: $|\vec{C}| = \sqrt{c_1^2 + c_2^2 + c_3^2}$

3. Normalize the cross product to make it a unit vector: $\hat{C} = \frac{\vec{C}}{|\vec{C}|}$ This $\hat{C}$ is the unit vector perpendicular to both $\vec{A}$ and $\vec{B}$.

Please provide the components of the two vectors if you need a specific solution.

Would you like to proceed with this, or do you have any other questions?

Here are some related questions for further exploration:

1. What properties does the cross product vector have with respect to the original vectors?
2. How would you find a vector perpendicular to only one given vector?
3. What if the vectors $\vec{A}$ and $\vec{B}$ are parallel? What would the cross product be?
4. Can the magnitude of a cross product ever be negative?
5. How does the cross product differ in 2D and 3D spaces?

Tip: Remember that the direction of the cross product follows the right-hand rule, which can help determine its orientation in space.