Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR ALGEBRAPp − apdAbθaOFigure 7.14The minimum distance from a point to a line.◮Find the minimum distance from the point P with coordinates (1, 2, 1) to the line r = a+λb,where a = i + j + k and b =2i − j +3k.Comparison with (7.39) shows that the line passes through the point (1, 1, 1) and hasdirection 2i − j +3k. The unit vector in this direction isˆb = √ 1 (2i − j +3k).14The position vector of P is p = i +2j + k and we find(p − a) × ˆb = √ 1 [ j × (2i − 3j +3k)]14= √ 1 (3i − 2k).14Thus the minimum distance from the line to the point P is d = √ 13/14. ◭7.8.2 Distance from a point to a planeThe minimum distance d from a point P whose position vector is p to the planedefined by (r − a) · ˆn = 0 may be deduced by finding any vector from P to theplane and then determining its component in the normal direction. This is shownin figure 7.15. Consider the vector a − p, which is a particular vector from P tothe plane. Its component normal to the plane, and hence its distance from theplane, is given byd =(a − p) · ˆn, (7.46)where the sign of d depends on which side of the plane P is situated.230