To determine the value(s) of $t$ for which the given points lie in the same plane, we need to ensure that these points are coplanar.

Steps:

1. Define the points: $A = (t, 1, -2), \quad B = (2, t, -1), \quad C = (-1, 2, t), \quad D = (t, 8, -3)$

2. Condition for coplanarity: Points are coplanar if the scalar triple product of three vectors formed by these points is zero. The vectors can be $\vec{AB}, \vec{AC}, \vec{AD}$.

3. Find vectors: $\vec{AB} = (2-t, t-1, 1), \quad \vec{AC} = (-1-t, 1, t+2), \quad \vec{AD} = (0, 7, -1)$

4. Compute the scalar triple product: The scalar triple product is given by: $\vec{AB} \cdot (\vec{AC} \times \vec{AD}) = 0$ Compute $\vec{AC} \times \vec{AD}$, then dot it with $\vec{AB}$.

Let me calculate this for you.import sympy as sp

Define variables and vectors

t = sp.symbols('t') AB = sp.Matrix([2 - t, t - 1, 1]) AC = sp.Matrix([-1 - t, 1, t + 2]) AD = sp.Matrix([0, 7, -1])

Compute cross product AC x AD

cross_product = AC.cross(AD)

Compute scalar triple product AB . (AC x AD)

scalar_triple_product = AB.dot(cross_product)

Solve for t where scalar triple product is zero

t_values = sp.solve(scalar_triple_product, t) t_values[-2, 3][-2, 3]successThe values of $t$ for which the points lie in the same plane are:

$t = -2 \quad \text{and} \quad t = 3$

Let me know if you'd like me to explain this process further or provide additional details!

Related Questions:

1. What is the geometric interpretation of the scalar triple product?
2. How can you verify coplanarity of points using a determinant?
3. Can the method be extended to higher dimensions?
4. How does the condition for coplanarity change if one point is at the origin?
5. What are some practical applications of coplanarity in physics or engineering?

Tip:

Always visualize the vectors in 3D space when solving coplanarity problems to better understand their relationships.