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To find a unit vector that is orthogonal to both vectors \(\vec{u} = (6, 2, 8)\) and \(\vec{v} = (4, -3, -1)\), we can follow these steps:

### Step 1: Find the Cross Product of \(\vec{u}\) and \(\vec{v}\)
The cross product \(\vec{u} \times \vec{v}\) gives a vector that is orthogonal to both \(\vec{u}\) and \(\vec{v}\).

\[
\vec{u} \times \vec{v} = 
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
6 & 2 & 8 \\
4 & -3 & -1 \\
\end{vmatrix}
\]

This determinant expands to:

\[
\vec{u} \times \vec{v} = \hat{i}(2 \cdot (-1) - 8 \cdot (-3)) - \hat{j}(6 \cdot (-1) - 8 \cdot 4) + \hat{k}(6 \cdot (-3) - 2 \cdot 4)
\]

Calculating each component:

1. **i-component**:
   \[
   2 \cdot (-1) - 8 \cdot (-3) = -2 + 24 = 22
   \]

2. **j-component**:
   \[
   -(6 \cdot (-1) - 8 \cdot 4) = -(-6 - 32) = 38
   \]

3. **k-component**:
   \[
   6 \cdot (-3) - 2 \cdot 4 = -18 - 8 = -26
   \]

So,

\[
\vec{u} \times \vec{v} = (22, 38, -26)
\]

### Step 2: Normalize the Vector
To make this vector a unit vector, we need to divide it by its magnitude.

1. **Magnitude of \((22, 38, -26)\)**:
   \[
   \sqrt{22^2 + 38^2 + (-26)^2} = \sqrt{484 + 1444 + 676} = \sqrt{2604} = 51.0235
   \]

2. **Unit Vector**:
   \[
   \text{Unit vector} = \frac{1}{51.0235} \times (22, 38, -26) \approx (0.43147, 0.74497, -0.50934)
   \]

### Final Answer
The unit vector that is orthogonal to both \(\vec{u}\) and \(\vec{v}\) is approximately:

\[
\boxed{(0.43147, 0.74497, -0.50934)}
\]

Would you like further clarification, or do you have any questions on this solution?

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### Follow-Up Questions:
1. What is the significance of the cross product in vector mathematics?
2. How do you calculate the magnitude of a vector?
3. Why is it important to normalize vectors in certain applications?
4. Can there be more than one unit vector orthogonal to two given vectors?
5. How does rounding affect the accuracy of vector calculations?

### Tip:
For higher precision in calculations, avoid rounding intermediate steps too early, especially in sensitive problems.

Find a unit vector that is orthogonal to both \( \vec{u} = (6, 2, 8) \) and \( \vec{v} = (4, -3, -1) \).

Learn how to find a unit vector orthogonal to both vectors \( \vec{u} = (6, 2, 8) \) and \( \vec{v} = (4, -3, -1) \) through cross product and normalization. This problem demonstrates essential vector concepts like cross product, magnitude, and unit vectors. Ideal for advanced high school or college-level students.

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