Mathematics

Mathematics

Problem

The perpendicular distance of a corner of a unit cube from a diagonal not passing through it is a. 32\sqrt{\frac{3}{2}} b. 23\sqrt{\frac{2}{3}} c. 34\sqrt{\frac{3}{4}} d. 43\sqrt{\frac{4}{3}}
Intuitive explanation

A unit cube has four main diagonals connecting opposite corners. The key insight is to use the point-to-line distance formula in 3D: pick any corner not on your chosen diagonal, set up coordinates for the diagonal as a line in space, then find the perpendicular distance. Symmetry helps—all such corners are equidistant from a given diagonal, so you only need to calculate once using vectors and the cross product.

Intuitive explanation

A unit cube has four main diagonals connecting opposite corners. The key insight is to use the point-to-line distance formula in 3D: pick any corner not on your chosen diagonal, set up coordinates for the diagonal as a line in space, then find the perpendicular distance. Symmetry helps—all such corners are equidistant from a given diagonal, so you only need to calculate once using vectors and the cross product.

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