Vector Equation Of Line

Question

The shortest distance between two skew lines is computed using which expression involving their direction vectors and a connecting vector between points on them?

RankGuru · Accepted Answer

For two skew lines r = a1 + s b1 and r = a2 + t b2, the shortest distance equals the length of the common perpendicular, given by |(b1 x b2) . (a2 - a1)| / |b1 x b2|. This formula combines the scalar and cross products and is a central JEE Advanced result in three-dimensional vector geometry. The cross product b1 x b2 is perpendicular to both lines, defining the direction of the common perpendicular; projecting the connecting vector (a2 - a1) onto this direction, then normalizing by the cross product's magnitude, yields the gap between the lines. Option two replaces the cross product with a dot product in the numerator and misuses a sum in the denominator, breaking the perpendicularity logic. Option three adds vectors where a scalar projection is required, which is dimensionally inconsistent. Option four projects the connecting vector onto itself, ignoring the line directions entirely. The numerator is precisely the scalar triple product [b1, b2, (a2 - a1)] in absolute value. Plausibility check: if the lines intersect, the connecting vector lies in the plane of b1 and b2, making the triple product zero, so the formula correctly returns a distance of zero.

Vector Equation Of Line

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