Understanding Engineering Mathematics

Exercise on 11.5

Calculate the distances of the points (i) (1, 0, 2), (ii) (−2, 1,−3), (iii) (−1,−1,−4) from
the origin. Also calculate the distance between each pair of points.

Answer

(i)

√

5 (ii)

√

14 (iii)

√
18
Between (i) and (ii),

√

35. Between (i) and (iii),

√

41. Between (ii) and (iii),

√

6.

11.6 Direction cosines and ratios

NB. Sections 11.6 – 8 are not essential to what follows and may be safely omitted for
the student who simply wants to be able to use vectors. However, the ideas are very
important in more advanced engineering mathematics and occur in such topics as CAD
and surveying.
While the coordinates of a pointP(x,y,z), tell us everything that we need to know
about the line segmentOP, they are not very convenient for indicating its direction, which
is most easily done by means of angles. When soldiers aim a field gun they do not specify
the coordinates of the tip of the barrel – they give the angle of elevation of the barrel,
and the bearing of the target. Similarly, the best way to specify the direction ofOPis
to specify the angles it makes with the axes. But we still want to retain a link with the
coordinates(x, y, z)and for this reason we use the cosines of these angles – the so-called
direction cosines.
LetOPmake anglesα,β,γwith the axesOx,Oy,Ozrespectively – see Figure 11.7.

P

PN is perpendicular to Ox

ON = |x| = r |cos a|

r = |OP|

0

N

y

x

a

g b

z

Figure 11.7Direction angles in three dimensions.

Thedirection cosinesof the line segment, orradius vector,OPare defined as cosα,
cosβ,cosγdenoted by:

l=cosα, m=cosβ, n=cosγ

Since x=rcosα, y=rcosβ, z=rcosγ

we have
l=

x

r

,m=

y

r

,n=

z

r