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

Select the correct option:

🔒 Solution Hidden from View

More vector algebra Practice Questions

Two vectors of magnitudes 5 and 12 act at a point with an included angle of 90 degrees; evaluate the...

If the scalar triple product [a b c] equals 7 for three given vectors, what is the value of the rear...

Three unit vectors a, b and c satisfy a + b + c = 0; using this constraint, evaluate the sum a . b +...

For two vectors with magnitudes 4 and 3 enclosing a 30 degree angle, evaluate the sum of squares |a ...

Three vectors a = i + 2j + 3k, b = 2i + j + k and c = 3i + j + 2k are given; compute their scalar tr...

If two nonzero vectors a and b satisfy |a + b| = |a - b|, what can be concluded about the angle betw...

Consider the vectors a = 3i + 4j and b = i + 2j + 2k; find the scalar projection of vector a onto th...

What is the unit vector directed along the vector obtained by adding a = 2i - j + 2k and b = i + j -...

For what value of the scalar lambda are the three vectors i + j + k, i + lambda j + 2k and 2i + j + ...

The position vectors of three points are A(1,1,1), B(2,3,5) and C(3,5,1); find the area of triangle ...

Using the standard expansion identity, simplify the vector triple product a x (b x c) where a, b and...

A point P divides the line segment joining A(1,2,3) and B(4,8,9) internally in the ratio 2:1; locate...

About This Question

Solution