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Consider a cube Q defined by the vertices {(x1, x2, x3) ∈ R³: x1, x2, x3 ∈ {0, 1}}. Let F represent the collection of all twelve lines formed by the diagonals of the six faces of the cube. Additionally, let S denote the group of four lines that pass through the main diagonals of the cube, such as the line connecting (0,0,0) and (1,1,1). If d(ℓ1, ℓ2) represents the shortest distance between two lines ℓ1 and ℓ2, determine the maximum value of d(ℓ1, ℓ2) when ℓ1 is chosen from F and ℓ2 is chosen from S.

1. 1/√6
2. 1/√8
3. 1/√3
4. 1/√12

Correct answer: 1/√6

Solution

The maximum distance between a face diagonal and a main diagonal of the cube is determined geometrically. Using the properties of the cube and the shortest distance formula for skew lines, the maximum value is found to be 1/√6.

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