The problem asks us to find the value of $k$ such that the vectors $\vec{a} = (2, 3, 1)$, $\vec{b} = (1, 2, 3)$, and $\vec{c} = (8, 5, k)$ all lie on the same plane.

To determine if three vectors lie on the same plane, their scalar triple product must equal zero. The scalar triple product is given by:

$\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$

Let's first compute the cross product $\vec{b} \times \vec{c}$, and then take the dot product of $\vec{a}$ with the result.

Cross product: $\vec{b} \times \vec{c}$

The vectors $\vec{b} = (1, 2, 3)$ and $\vec{c} = (8, 5, k)$. The formula for the cross product is:

$\vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 8 & 5 & k \end{vmatrix}$

Let's compute this step by step.The cross product of vectors $\vec{b} = (1, 2, 3)$ and $\vec{c} = (8, 5, k)$ is:

$\vec{b} \times \vec{c} = (-13, 23, -11)$

Next, we compute the dot product of $\vec{a} = (2, 3, 1)$ with the result of $\vec{b} \times \vec{c}$.

$\vec{a} \cdot (\vec{b} \times \vec{c}) = (2, 3, 1) \cdot (-13, 23, -11)$

Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, let me know if you'd like to see the next steps explained in detail or have any specific questions about this process.

Here are five related questions you might want to explore:

1. How do we interpret the scalar triple product geometrically?
2. What is the significance of the cross product in physics and engineering?
3. How can we find the area of a parallelogram formed by two vectors using the cross product?
4. What happens if three vectors do not lie on the same plane? What does that mean geometrically?
5. How does the determinant of a matrix relate to the scalar triple product?

Tip: The scalar triple product can be interpreted as the volume of the parallelepiped formed by three vectors. If this volume is zero, the vectors are coplanar.