To determine the value of $k$ for which the vectors are linearly dependent, we need to check when the determinant of the matrix formed by these vectors is zero.

Given: $\mathbf{v}_1 = \begin{pmatrix} p \\ 0 \\ 2 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} k \\ 4 \\ 2 \end{pmatrix}$

and $p = 3$, the vectors are:

$\mathbf{v}_1 = \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} k \\ 4 \\ 2 \end{pmatrix}$

These vectors form the columns of a matrix:

$A = \begin{pmatrix} 3 & 0 & k \\ 0 & 2 & 4 \\ 2 & 4 & 2 \end{pmatrix}$

We need to find the value of $k$ such that the determinant of matrix $A$ is zero.

$\text{det}(A) = \begin{vmatrix} 3 & 0 & k \\ 0 & 2 & 4 \\ 2 & 4 & 2 \end{vmatrix}$

Let's compute this determinant.The determinant of matrix $A$ is given by:

$\text{det}(A) = -4k - 36$

For the vectors to be linearly dependent, this determinant must be zero:

$-4k - 36 = 0$

Let's solve for $k$.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can still help you solve this problem manually.

We have the determinant equation:

$-4k - 36 = 0$

To find the value of $k$, we solve for $k$:

$-4k = 36$

$k = -9$

So, the value of $k$ that makes the vectors linearly dependent is $k = -9$.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How can we verify that the vectors are linearly dependent?
2. What is the geometric interpretation of linearly dependent vectors?
3. How do you determine linear independence for more than three vectors?
4. What happens when the determinant of a matrix is zero?
5. How do you find a basis for the span of these vectors?

Tip: Remember that a set of vectors is linearly dependent if one vector can be written as a linear combination of the others.