Step 1: Problem Definition
Given four points in 3D space: A(4, -2, 1), B(7, -4, 7), C(2, -5, 10), and D(-1, -3, 4). Determine the geometric figure formed by these points.
Step 2: Vector Calculation
Compute the vectors representing the sides:\[\vec{AB} = B - A = (7-4, -4-(-2), 7-1) = (3, -2, 6),\]\[\vec{AD} = D - A = (-1-4, -3-(-2), 4-1) = (-5, -1, 3),\]\[\vec{BC} = C - B = (2-7, -5-(-4), 10-7) = (-5, -1, 3),\]\[\vec{DC} = C - D = (2-(-1), -5-(-3), 10-4) = (3, -2, 6).\]
Step 3: Parallelogram Verification
A quadrilateral is a parallelogram if its opposite sides are equal and parallel.
Observation: \( \vec{AB} = \vec{DC} \) and \( \vec{AD} = \vec{BC} \). This confirms that opposite sides are equal and parallel.
Step 4: Exclusion of Other Figures
Tetrahedron: Requires four triangular faces; not applicable here.
Rhombus: Requires all sides to be equal; not satisfied by current calculations.
Square: Requires all sides to be equal and all angles to be 90 degrees; not satisfied.
Step 5: Conclusion Based on Options
The figure formed by the given points is a parallelogram, corresponding to option (B).Final Answer: The points form a (B) Parallelogram.