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

### Step 1: Find the Cross Product
The cross product \( \mathbf{u} \times \mathbf{v} \) will give a vector that is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). The cross product of \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \) is given by:

\[
\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}
\]

Substitute \( \mathbf{u} = (6, 2, 8) \) and \( \mathbf{v} = (4, -3, -1) \):

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

Expanding the determinant:

\[
= \mathbf{i} (2 \cdot -1 - 8 \cdot -3) - \mathbf{j} (6 \cdot -1 - 8 \cdot 4) + \mathbf{k} (6 \cdot -3 - 2 \cdot 4)
\]

Calculating each term:

- For \( \mathbf{i} \)-component: \( 2 \cdot -1 - 8 \cdot -3 = -2 + 24 = 22 \)
- For \( \mathbf{j} \)-component: \( -(6 \cdot -1 - 8 \cdot 4) = -(-6 - 32) = 38 \)
- For \( \mathbf{k} \)-component: \( 6 \cdot -3 - 2 \cdot 4 = -18 - 8 = -26 \)

So, \( \mathbf{u} \times \mathbf{v} = (22, 38, -26) \).

### Step 2: Convert to a Unit Vector
To make this vector a unit vector, we divide it by its magnitude. The magnitude of \( (22, 38, -26) \) is:

\[
\sqrt{22^2 + 38^2 + (-26)^2} = \sqrt{484 + 1444 + 676} = \sqrt{2604}
\]

\[
\sqrt{2604} \approx 51.0294
\]

Thus, the unit vector \( \mathbf{n} \) orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \) is:

\[
\mathbf{n} = \frac{1}{51.0294} (22, 38, -26) \approx \left( \frac{22}{51.0294}, \frac{38}{51.0294}, \frac{-26}{51.0294} \right)
\]

Calculating each component:

\[
\mathbf{n} \approx (0.43117, 0.74487, -0.50940)
\]

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

\[
\mathbf{n} \approx (0.43117, 0.74487, -0.50940)
\]

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Would you like a more detailed explanation of any part of this process? Here are some related questions for further exploration:

1. What are other methods to find a unit vector orthogonal to two vectors?
2. How does the cross product determine the direction of an orthogonal vector?
3. How is the magnitude of a vector calculated in three-dimensional space?
4. Why is the cross product important in physics and engineering?
5. How do rounding errors impact vector calculations, and how can they be minimized?

**Tip:** Always re-check your magnitude calculations for precision, especially in sensitive problems involving unit vectors!

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

Step-by-step solution to find a unit vector orthogonal to both vectors \( \mathbf{u} = (6, 2, 8) \) and \( \mathbf{v} = (4, -3, -1) \). Learn about cross product calculation, magnitude, and unit vector formulation. Suitable for advanced high school math students.

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