To determine the value of $z$ such that the points $(2, 1, z)$, $(-1, 2, -1)$, $(3, 0, 1)$, and $(1, -3, 4)$ lie in the same plane, we calculate the scalar triple product. This requires the determinant of a matrix formed by vectors derived from these points to be zero.

Steps:

1. Construct three vectors using the points: $\mathbf{v_1} = (-1 - 2, 2 - 1, -1 - z) = (-3, 1, -1 - z)$ $\mathbf{v_2} = (3 - 2, 0 - 1, 1 - z) = (1, -1, 1 - z)$ $\mathbf{v_3} = (1 - 2, -3 - 1, 4 - z) = (-1, -4, 4 - z)$

2. Form the determinant of the 3x3 matrix:

   \begin{vmatrix} -3 & 1 & -1 - z \\ 1 & -1 & 1 - z \\ -1 & -4 & 4 - z \end{vmatrix}$$

3. Expand the determinant:

   \begin{vmatrix} -1 & 1 - z \\ -4 & 4 - z \end{vmatrix} - 1 \begin{vmatrix} 1 & 1 - z \\ -1 & 4 - z \end{vmatrix} + (-1 - z) \begin{vmatrix} 1 & -1 \\ -1 & -4 \end{vmatrix}$$

4. Simplify each 2x2 determinant: $\mathbf{det}_1 = (-1)(4 - z) - (-4)(1 - z) = -4 + z + 4 - 4z = z - 4z + 4 = z - 4z + 4$ Similarly for sympy as sp

Define the variable z

z = sp.symbols('z')

Define the vectors from the points

v1 = sp.Matrix([-3, 1, -1 - z]) v2 = sp.Matrix([1, -1, 1 - z]) v3 = sp.Matrix([-1, -4, 4 - z])

Compute the determinant of the matrix formed by the vectors

determinant = sp.det(sp.Matrix([v1, v2, v3]).transpose())

Solve the determinant equation to find z

solution = sp.solve(determinant, z) solutionIt seems there was an issue due to an incorrect matrix shape. Let me correct the setup and compute the determinant properly. Correcting matrix shape to form the determinant properly

Combine the vectors into a square matrix (3x3)

matrix = sp.Matrix([ [-3, 1, -1 - z], [1, -1, 1 - z], [-1, -4, 4 - z] ])

Compute the determinant

determinant = matrix.det()

Solve the determinant equation to find z

solution = sp.solve(determinant, z) solutionThe value of $z$ such that the points $(2, 1, z)$, $(-1, 2, -1)$, $(3, 0, 1)$, and $(1, -3, 4)$ lie in the same plane is:

$z = 0$

Let me know if you want further clarification or details!

Here are some related questions you might explore:

1. How do we confirm the four points are coplanar using vectors?
2. What happens if one of the points is removed? Would the remaining three still define a plane?
3. How can the determinant be interpreted geometrically in this context?
4. Can this method be generalized for $n$-dimensional space?
5. What other methods exist to verify if points lie in the same plane?

Tip: When working with determinants and vectors in geometry, always double-check matrix dimensions to ensure accurate calculations!