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Mathematics-Online course: Vector Calculus - Lines

Distance of Two Lines

The distance between two lines and which are not parallel and with parametric equations

and

is given by

This follows from the geometric properties of the dot and the cross product because the distance coincides with the distance of from the plane containing and the by translated line

Using the parallelepidial product the distance may be expressed as

where and are the points with location vector and rsp.

For parallel lines we have

Two lines are called skew lines if they are not parallel and if the distance between them is positive.

Denoting the position vectors of the points of minimal distance by

then

is orthogonal to both direction vectors, and, consequently, is parallel to

Since the magnitude of a vector equals the absolute value of its scalar product with an parallel unit vector, it follows that

and we obtain the desired formula.

For parallel lines we can use the formula for the distance of a point from a line.

The distance between the lines

amounts to