Write a unit vector perpendicular to both 

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.

Write a unit vector perpendicular to both vectors A and B.

This guide provides a step-by-step solution for finding a unit vector perpendicular to two vectors using the cross product. It includes formulas for calculating the cross product, its magnitude, and how to normalize the result. Ideal for students in Grades 11-12 studying vector algebra.

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