Concepts: Shortest distance between two lines

4.1 Lines that meet, run parallel or stay apart

Two lines in space can meet at a point, run parallel, or pass each other without ever meeting. The last kind is called skew, and for skew lines there is a well defined shortest distance between the two lines. It is the length of the common perpendicular segment joining the two lines. For intersecting lines this distance is zero, and for parallel lines it is measured along a perpendicular between the two.

4.2 The skew-line formula

Write the two lines as r = a1 + lambda b1 and r = a2 + mu b2. The vector joining the two base points is a2 - a1. The common perpendicular direction is perpendicular to both b1 and b2, so it points along their cross product. The shortest distance is the projection of the joining vector on this perpendicular direction, which is the scalar triple product divided by the magnitude of the cross product.

Three checks follow from the same formula. If the numerator is zero, the joining vector is perpendicular to the common normal, which means the two lines lie in one plane; they may be intersecting or parallel, and the distance is zero. If the cross product is zero, the directions are parallel and the formula must be replaced by the parallel-line version. If neither holds, the lines are skew and the distance is positive.

4.3 The parallel-line case

Parallel lines have the same direction vector. The distance between them is the component of the joining vector perpendicular to that direction. In practice, take the difference of the two base points, cross it with the shared direction, and divide by the length of the direction. The magnitude of the cross product is the area of the parallelogram and the denominator turns that area into a height.

A special case is worth remembering. If one line passes through the origin, the distance from the second line to it can be computed from the same formula with one base point at zero. This is the version that appears in questions asking for the distance of a line from the origin.

The formula can also be read as a statement about heights. The numerator chooses how much of the joining vector is along the common perpendicular, and the denominator fixes the unit of that direction, so the distance never depends on which points on the two lines are chosen as base points. That is why any convenient pair of points may be used. A useful special case is when one line passes through the origin, because then one base point is the zero vector and the computation simplifies.

The cartesian form

When both lines are written in cartesian form, the same computation is done with direction ratios. The cross product of the two direction triples gives the common normal, the difference of the two points gives the joining vector, and the triple product is a three by three determinant. Boards accept either the vector route or the determinant route, so choose whichever is faster for the given data.

4.4 Worked examples

Example 1. One line has base point (1, -2, 3) and direction (2, 1, -1); the other has base point (2, -1, 1) and direction (1, 3, -2). Their cross product is (1, 3, 5), and the vector joining the base points is (1, 1, -2). The triple product is 1 + 3 - 10, whose modulus is 6, and the cross product has magnitude root 14, so the distance is 6 divided by root 14.

Example 2. Find the distance between the parallel lines r = (i + j) + lambda(i + 2 j - k) and r = (2 i - j + k) + mu(i + 2 j - k). The joining vector is (1, -2, 1) and the shared direction is (1, 2, -1). The cross product is (0, 2, 4), with magnitude 2 root 5, and the direction length is root 6, so the distance is 2 root 5 divided by root 6.

Example 3. Find the distance of the line (x + 3)/5 = (y - 1)/2 = (z + 4)/3 from the point (0, 2, 3) using the cross-product method, and find the foot of the perpendicular. The direction is (5, 2, 3) and the vector from the line point to the given point is (3, 1, 7), so the distance is the magnitude of the cross product divided by the direction length. Computed directly, the foot is (2, 3, -1) and the distance is root 21.

Example 4. Two lines have direction ratios (2, 3, 4) and (5, 2, 1), and pass through points (1, 2, 3) and (4, 1, 0). The cross product of the directions is (-5, 18, -11), and the joining vector is (3, -1, -3). Its triple product with the two directions is non-zero, so the lines are skew and the shortest distance is positive.