1. A Line makes angles 
and 
with the positive axes of x and y respectively. Find the angle that it makes with the positive axis of z.
Sol: Given: α= β =
Required Angle: γ
2. If (3,-4,12) are direction ratios of a line then find the direction cosines of the line.
Sol:
Given: a = 3 | b = -4|, c = 12
3. Find the direction ratios and direction cosines of the line joining the points (5,-2,3) and
(-2,3,7)
Sol: Given Points: A = (5,-2,3)
B = (-2,3,7)
4. Find the angle between the lines whose direction ratios are (1, −2, 1) and (−1, 1, 0)
Sol:
Given d.r.'s: (a1,b1,c1) = (1,−2,1) (a2,b2,c2) = (−1,1,0)
θ = 150°
Angles between the given lines are 30°, 150°
5. Find the equations of the line passing through (2,−3,1) and having direction ratios
(3,2,−5)
sol:
6. Find the equations of the line passing through (−2, 3, −1) and (3, 4, 2)
Sol:
Given Points: (x1, y1, z1) = (−2, 3, −1)
(x2, y2, z2) = (3, 4, 2)
7. Find the coordinates of the foot of the perpendicular drawn from A(1,0,3) to the line joining the points B (4,7,1) and C (3,5,3)
Sol: Foot of the perpendicular means the points of intersection of two perpendicular lines.
Let foot of the perpendicular : D
Let D divide 
in the ratio λ : 1
⇒ 2λ + 3 + 10λ + 14 + 4 = 0
⇒ 12λ + 21 = 0
∴ Foot of the Perpendicular :
⇒ 
⇒ 
8. Find the length of the projection of the join of P (3, 4, 5) and Q( 4, 6, 3) on the line joining A (−1, 2,4 ) and B (1, 0, 5)
sol: Given Points: A (−1, 2, 4), B (1, 0, 5)
P (3, 4, 5), Q (4, 6, 3)
D.r.s of : (1+1, 0−2, 5−4)
⇒ (2,−2,1)
⇒ 
∴ Projection of the lin e segment join of P (3, 4, 5) and Q (4, 6, 3) on
: = | l (X2 - X1) + m (y2 - y1) + n (z2 - z1) |
9. Find the angle between the lines whose d.c.s are given by the equations 3l + m + 5n = 0
and 6mn − 2nl + 5lm = 0
Sol: Given Lines: 3l + m + 5n = 0 −−− (1)
⇒ m = − (3l + 5n)
6mn − 2nl + 5lm = 0 −−− (2)
from (1) and (2)
−6n(3l + 5n) − 2nl − 5l(3l + 5n) = 0
⇒ − 18nl − 30n2 − 2nl − 15l2 − 25nl = 0
⇒ −15l 2 − 45nl − 30n2 =0
⇒ l2 + 3nl + 2n2 = 0
l2 + 2nl + nl + 2n2 = 0
⇒ l(l + 2n) + n(l + 2n) = 0
⇒ (l + n)(l + 2n) = 0
⇒ l + n = 0 −−− (3) l + 2n = 0 −−− (4)
Solving (1) and (3)
3l + m + 5n = 0
l + 0.m + n = 0
D.r.s of one Line : (1, 2, −1)
Solving (1) and (4)
D.rs of Second Line: (2, −1, −1)
Let 'θ' be the angle between the lines then
10. If a line makes angles α, β, γ, δ with the four diagonals of a Cube then prove that cos 2 α+ cos2 β + cos2 γ + cos2 δ = 
Sol:
Let the length of an edge of the Cube be 1
Let one corner of the cube be taken as the origin and the three edges through this corner as
Direction Cosines and Direction Ratios.. (New Syllabus) >> Page - 14
the axes from the figure,
A = (1, 0, 0)
B = (0, 1, 0)
C = (0, 0, 1)
P = (1, 1, 1)
Q = (1, 1, 0)
R = (0, 1, 1)
S = (1, 0, 1)
Let (l, m, n) be the D.c.s of the given line and α, β, γ, δ be its inclinations with the diagonals.
Then
cos2 α + cos2 β + cos2 γ + cos2 δ =
[ (l + m + n) 2 + ( -l + m + n ) 2 + ( l - m + n ) 2 + ( l + m - n) 2 ]
= [ l 2 + m2 + n2 ]
= (1) [ ... l 2 + m2 + n2 = 1]
=
cos2 α + cos2 β + cos2 γ + cos2 δ =
