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The problem asks to find the value of \( k \) such that the vectors \( \vec{a} = \langle 2, 3, 1 \rangle \), \( \vec{b} = \langle 1, 2, 3 \rangle \), and \( \vec{c} = \langle 18, 5, k \rangle \) all lie on the same plane.

To check if three vectors lie on the same plane, we can check if the scalar triple product of these vectors equals zero. The scalar triple product of vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) is given by the determinant:

\[
\vec{a} \cdot (\vec{b} \times \vec{c}) = \text{det} \begin{pmatrix}
2 & 3 & 1 \\
1 & 2 & 3 \\
18 & 5 & k
\end{pmatrix}
\]

Let me calculate this determinant to find the value of \( k \).The determinant of the matrix with \( k = 101 \) is approximately 202, which is not equal to zero. Thus, \( k = 101 \) does not satisfy the condition for the vectors to lie on the same plane.

I will now test the other values of \( k \) from the options.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the steps to manually compute the determinant and find the value of \( k \).

Would you like a detailed explanation of the process? 

Here are five related questions that expand on this topic:
1. How do you calculate the cross product of two vectors?
2. What does it mean for vectors to be coplanar?
3. How do you compute the determinant of a 3x3 matrix manually?
4. Why is the scalar triple product useful in vector analysis?
5. Can the scalar triple product be used to find the area or volume of geometric figures?

**Tip:** To check if three vectors are coplanar, always remember that their scalar triple product must be zero.

Let vectors a = (2, 3, 1), b = (1, 2, 3), and c = (18, 5, k). Find the value of k such that the vectors a, b, and c all lie on the same plane.

Determine the value of k for which the vectors a = (2, 3, 1), b = (1, 2, 3), and c = (18, 5, k) are coplanar. This problem involves the use of the scalar triple product and the determinant of a 3x3 matrix. Suitable for students in Grades 10-12.

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