To find the value(s) of $k$ such that the vectors $\mathbf{u} = [6, 1, 8]$ and $\mathbf{v} = [k, -4, 5]$ are orthogonal, we use the condition that two vectors are orthogonal if their dot product is zero.

The dot product of two vectors $\mathbf{u} = [u_1, u_2, u_3]$ and $\mathbf{v} = [v_1, v_2, v_3]$ is given by:

$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$

For the vectors $\mathbf{u} = [6, 1, 8]$ and $\mathbf{v} = [k, -4, 5]$, we have:

$6k + 1(-4) + 8(5) = 0$

Simplifying this expression:

$6k - 4 + 40 = 0$ $6k + 36 = 0$ $6k = -36$ $k = -6$

Thus, the value of $k$ is $\boxed{-6}$.

Would you like further details on this solution or any additional explanation?

Here are 5 related questions for deeper understanding:

1. What does it mean for two vectors to be orthogonal?
2. How do you compute the dot product of two vectors?
3. How would the solution change if the vectors were not orthogonal?
4. Can you give an example of two 2D vectors that are orthogonal?
5. How does the concept of orthogonality relate to linear independence?

Tip: The dot product is zero when vectors are perpendicular—an essential property in vector geometry and linear algebra!