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The problem asks for two unit vectors that are orthogonal to both \( \vec{u} = \langle 4, 4, -4 \rangle \) and \( \vec{v} = \langle 0, 5, 3 \rangle \).

To find the vectors that are orthogonal to both \( \vec{u} \) and \( \vec{v} \), we need to compute their cross product first. The cross product of two vectors yields a vector that is orthogonal to both.

Let's start by computing the cross product of \( \vec{u} \) and \( \vec{v} \):

\[
\vec{w} = \vec{u} \times \vec{v}
= \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
4 & 4 & -4 \\
0 & 5 & 3
\end{vmatrix}
\]

Next, we'll calculate this cross product step by step.The cross product of \( \vec{u} \) and \( \vec{v} \) is:

\[
\vec{w} = \langle 32, -12, 20 \rangle
\]

This vector is orthogonal to both \( \vec{u} \) and \( \vec{v} \). However, we are asked for unit vectors, so we need to normalize this vector by dividing it by its magnitude.

The magnitude of \( \vec{w} \) is:

\[
|\vec{w}| = \sqrt{32^2 + (-12)^2 + 20^2}
\]

Let's calculate the magnitude and then find the unit vector.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, I can guide you through the process to manually calculate the magnitude and normalize the vector \( \vec{w} = \langle 32, -12, 20 \rangle \).

1. **Find the magnitude of \( \vec{w} \):**
   \[
   |\vec{w}| = \sqrt{32^2 + (-12)^2 + 20^2} = \sqrt{1024 + 144 + 400} = \sqrt{1568}
   \]

2. **Normalize \( \vec{w} \):**
   The unit vector is given by:
   \[
   \hat{w} = \frac{1}{|\vec{w}|} \langle 32, -12, 20 \rangle
   \]
   This will give you the unit vector.

Let me know if you would like me to explain any of these steps in detail!

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Here are 5 related questions you might find useful:

1. What are the properties of the cross product?
2. How do you verify if a vector is orthogonal to others?
3. What are applications of unit vectors in physics?
4. How do you find a unit vector in general?
5. What is the geometric interpretation of the magnitude of a vector?

**Tip:** When working with cross products, remember that the result is always orthogonal to both original vectors!

Find two unit vectors orthogonal to both \( \vec{u} = \langle 4, 4, -4 \rangle \) and \( \vec{v} = \langle 0, 5, 3 \rangle \). Give exact values (no decimals). Separate the vectors with a comma and do not pull any numbers or roots outside of the vector.

Discover how to find two unit vectors orthogonal to both \( \vec{u} = \langle 4, 4, -4 \rangle \) and \( \vec{v} = \langle 0, 5, 3 \rangle \) by computing their cross product and normalizing the result. This problem involves key concepts of linear algebra and vector calculus.

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