Draw a
perpendicular from P to QS to meet at N Now, in right angle triangle PNQ, ∠PQN=
γ  (Fig 11 3 (b) Therefore,
cosγ =
2
1
NQ
PQ
zPQ
−z
=
Similarly
cosα =
2
1
2
1
and cos
PQ
PQ
x
x
y
y
−
−
β =
Hence, the direction cosines of the line segment joining the points P(x1, y1, z1) and
Q(x2, y2, z2) are
2
1
PQ
x
−x
, 
2
1
PQ
y
−y
, 
2
1
PQ
z
z
−
where
PQ =
(
)
2
2
2
2
1
2
1
2
1
(
)
(
)
x
x
y
y
z
z
−
+ −
+ −
�Note   The direction ratios of the line segment joining P(x1, y1, z1) and Q(x2, y2, z2)
may be taken as
x2 – x1, y2 – y1, z2 – z1 or x1 – x2, y1 – y2, z1 – z2
Example 1 If a line makes angle 90°, 60° and 30° with the positive direction of x, y and
z-axis respectively, find its direction cosines