Understanding Engineering Mathematics

from which, since r^2 =x^2 +y^2 +z^2 ,

l^2 +m^2 +n^2 = 1

Note that the direction cosines of a line not passing through the origin are the same as
those of a parallel line through origin.
The numbersl,m,nare not always the most convenient to use – they are all less than
or equal to 1 in magnitude. Often all we need is simply the relation between them. Any
three numbersa,b,csatisfying (14
➤
)

a:b:c=l:m:n

are calleddirection ratiosofOP.

From l^2 +m^2 +n^2 =1wehave

l=

a

d

,m=

b

d

,n=

c

d

where
d=±

√

a^2 +b^2 +c^2

The±denotes that to a given set of direction ratios there are two sets of direction cosines
corresponding to oppositely directed parallel lines.

Problem 11.2

Calculate the direction cosines of each line to a vertex of a unit cube in

the first octant with the lowest corner at the origin (Figure 11.8).

(1,0,0) (1,1,0)

(1,1,1)

(0,1,1)

(0,0,1)

(0,1,0)

(0,0,0)

(1,0,1)

x

y

z

Figure 11.8

First note that the direction cosines of the origin are not defined, sincer=0 at the origin.
Consider the points on the axes – in this case the direction cosines are simply given by
the points themselves (1, 0, 0), (0, 1, 0), (0, 0, 1).
Now consider the point (1, 0, 1). The position vector to this makes an angle 45°with
thex-andz-axes, and 90°with theOy-axis. The direction cosines are therefore directly