Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR ALGEBRA

O

Q

P

ˆn

q

a

b

p

Figure 7.16 The minimum distance from one line to another.

Ifpandqare the position vectors of any two pointsPandQon different lines

then the vector connecting them isp−q. Thus, the minimum distancedbetween

the lines is this vector’s component along the unit normal, i.e.

d=|(p−q)·ˆn|.

A line is inclined at equal angles to thex-,y- andz-axes and passes through the origin.

Another line passes through the points(1, 2 ,4)and(0, 0 ,1). Find the minimum distance

between the two lines.

The first line is given by

r 1 =λ(i+j+k),

and the second by

r 2 =k+μ(i+2j+3k).

Hence a vector normal to both lines is

n=(i+j+k)×(i+2j+3k)=i− 2 j+k,

and the unit normal is

ˆn=

1

√

6

(i− 2 j+k).

A vector between the two lines is, for example, the one connecting the points (0, 0 ,0)
and (0, 0 ,1), which is simplyk. Thus it follows that the minimum distance between the
two lines is

d=

1

√

6

|k·(i− 2 j+k)|=

1

√

6

.

7.8.4 Distance from a line to a plane

Let us consider the liner=a+λb. This line will intersect any plane to which it

is not parallel. Thus, if a plane has a normalnˆthen the minimum distance from