Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR ALGEBRAQbqˆnPpaOFigure 7.16The minimum distance from one line to another.If p and q are the position vectors of any two points P and Q on different linesthen the vector connecting them is p − q. Thus, the minimum distance d betweenthe lines is this vector’s component along the unit normal, i.e.d = |(p − q) · ˆn|.◮A line is inclined at equal angles to the x-, y- and z-axes and passes through the origin.Another line passes through the points (1, 2, 4) and (0, 0, 1). Find the minimum distancebetween the two lines.The first line is given byr 1 = λ(i + j + k),and the second byr 2 = k + µ(i +2j +3k).Hence a vector normal to both lines isn =(i + j + k) × (i +2j +3k) =i − 2j + k,and the unit normal isˆn = √ 1 (i − 2j + k).6A vector between the two lines is, for example, the one connecting the points (0, 0, 0)and (0, 0, 1), which is simply k. Thus it follows that the minimum distance between thetwo lines isd = √ 1 |k · (i − 2j + k)| = √ 1 . ◭6 67.8.4 Distance from a line to a planeLet us consider the line r = a + λb. This line will intersect any plane to which itis not parallel. Thus, if a plane has a normal ˆn then the minimum distance from232