Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR ALGEBRA

O

A

P

θ

p−a d

a

b

p

Figure 7.14 The minimum distance from a point to a line.

Find the minimum distance from the pointPwith coordinates(1, 2 ,1)to the liner=a+λb,

wherea=i+j+kandb=2i−j+3k.

Comparison with (7.39) shows that the line passes through the point (1, 1 ,1) and has
direction 2i−j+3k. The unit vector in this direction is

ˆb=√^1

14

(2i−j+3k).

The position vector ofPisp=i+2j+kand we find

(p−a)×bˆ=

1

√

14

[j×(2i− 3 j+3k)]

=

1

√

14

(3i− 2 k).

Thus the minimum distance from the line to the pointPisd=

√

13 /14.

7.8.2 Distance from a point to a plane

The minimum distancedfrom a pointPwhose position vector ispto the plane

defined by (r−a)·nˆ= 0 may be deduced by finding any vector fromPto the

plane and then determining its component in the normal direction. This is shown

in figure 7.15. Consider the vectora−p, which is a particular vector fromPto

the plane. Its component normal to the plane, and hence its distance from the

plane, is given by

d=(a−p)·nˆ, (7.46)

where the sign ofddepends on which side of the planePis situated.