Direction Cosines and Direction Ratios: Topics
Direction Cosines and Direction Ratios: Topics
Direction Cosines and Direction Ratios: Topics
TOPICS:
1.DEFINITION OF D.CS., RELATION BETWEEN D.CS. OF A LINE, CO-ORDINATES OF A
POINT WHEN D.CS. ARE GIVEN AND DIRECTION COSINES OF A LINE JOINING TWO
POINTS.
2. ANGLE BETWEEN TWO LINES WHEN D.CS ARE GIVEN, FINDING THE ANGLE BETWEEN
TWO LINES WHEN THEIR D.CS ARE CONNECTED BY EQUATIONS.
5. CONDITIONS FOR PARALLEL AND PERPENDICULA LINES WHEN D.CS/D.RS ARE GIVEN.
6. ANGLE BETWEEN TWO LINES WHEN D.RS ARE GIVEN, FINDING THE ANGLE BETWEEN
TWO LINES WHEN THEIR D.RS ARE CONNECTED BY EQUATIONS.
DIRECTION COSINES & RATIOS (7 MARKS )
The angle between two skew lines is the angle between two lines drawn parallel to
them through any point in space.
DIRECTION COSINES
If α , β , γ are the angles made by a directed line segment with the positive
directions of the coordinate axes respectively, then cos a, cos b, cos g are called
the direction cosines of the given line and they are denoted by l, m, n respectively
Thus l = cos α , m = cos β, n = cos γ
The direction cosines of op are
If l, m, n are the d.c’s of a line L is one direction then the d.c’s of the same line in
the opposite direction are –l, –m, –n.
Note : The angles α, β, γ are known as the direction angles and satisfy the
condition 0 ≤ α, β, γ ≤ π .
Note : The sum of the angles α, β, γ is not equal to 2p because they do not lie in the
same plane.
Note: Direction cosines of coordinate axes.
π π
The direction cosines of the x-axis are cos 0, cos , cos i.e., 1, 0, 0
2 2
Similarly the direction cosines of the y-axis are (0,1,0) and z-axis are (0,0,1)
THEOREM
Note: If P(x,y,z) is any point in space such that OP =r then the direction cosines
of OP are x , y , z
r r r
Note: If P is any point in space such that OP =r and direction cosines of OP are
l,m,n then the point P =(lr,mr,nr)
Note: If P(x,y,z) is any point in space then the direction cosines of OP are
x y z
, ,
2+ 2+ 2+
x y +z
2 2
x y +z
2 2
x y2 + z2
THEOREM
If l, m, n are the direction cosines of a line L then l2 + m2 + n2 = 1.
Proof :
EXERCISE -6A
1. A line makes angle 900 , 600 and 300 with positive directions of x, y, z –axes
respectively. Find the direction cosines.
1 3
m = cos β = cos 600 = And n = cos γ = cos 300 =
2 2
Direction cosines of the line are 0, 1 , 3
2 2
⇒ a 2 + b2 + c 2 = 3 + 1 + 12 = 16 ⇒ a2 + b2 + c2 = 4
Direction cosines of OP are
a b c 3 1 2 3 3 1 3
, , = , , = , ,
a 2 + b2 + c2 a 2 + b2 + c2 a 2 + b 2 + c 2 4 4 4 4 4 2
4. Find the direction cosines of the line joining the points
( −4, 1, 7 ) and ( 2, − 3, 2)
Solution: A ( −4, 1, 2 ) and B ( 2, − 3, 2 ) are given points
d.rs of PQ are ( x2 − x1 , y2 − y1 , z2 − z1 )
i.e. (
2 + 4, 1 + 3, 2 − 7 ) i.e, ( 6, 4, − 5) = ( a, b, c )
⇒ a 2 + b 2 + c 2 = 36 + 16 + 25 = 77
Direction cosines of AB are
a b c 6 4 −5
, , = , ,
a + b2 + c2
2
a 2 + b2 + c2 2 2 2
a + b + c 77 77 77
II.
1. Find the direction cosines of the sides of the triangle whose vertices are
( 3, 5, − 4) , ( −1, 1, 2) and ( −5, − 5, − 2 )
Sol: A ( 3, 5, − 4 ) , B ( −1, 1, 2) and C ( −5, − 5, − 2) are the vertices of ∆ABC
d.rs of AB are ( −1 − 3, 1 − 5, 2 + 4) = ( − 4, − 4, 6)
Dividing with 16 + 16 + 36 = 68 = 2 17
i.e., −2 , −3 , −4
17 17 2 17
4 5 −1
Then d.cs of CA are , ,
42 42 42
2. Show that the lines PQ and RS are parallel where P, Q, R, S are two
points (2, 3, 4), (-1, -2, -1) and (1, 2, 5) respectively
III.
1. Find the direction cosines of two lines which are connected by the
relation l− 5m + 3n = 0 and 7l 2 + 5m2 −3n2 = 0
Sol. Given l − 5 m + 3n = 0
⇒ l = 5m − 3n − − − − − (1)
7 ( 5m − 3n ) + 5m 2 − 3n 2 = 0
2
⇒ 7 ( 25 2 + 9 n 2 − 30 mn ) + 5m 2 −3n 2 = 0
⇒ 6m2 − 7mn + 2n 2 = 0
⇒ ( 3m − 2n )( 2m − n ) = 0
m1 n1
Case (i): 3m1= 2n1 ⇒ =
2 3
2
Then m1 = n1
3
10
From (1) l1 = 5m1 − 3n1 = n1 − 3n1
3
10n1 − 9n1 n1
= =
3 3
l1 m1 n1
∴ = =
1 2 3
Dividing with 1 + 4 + 9 = 14
1 2 3
d.cs of the first line are , ,
14 14 14
⇒ l2 − 5m2 + 6m2 = 0
⇒ −l2 = m2
l2 m2 n2
∴ = =
−1 1 2
Dividing with 1 + 1 + 4 = 6
−1 1 2
d.cs of the second line are , ,
6 6 6
DIRECTION RATIOS
A set of three numbers a,b,c which are proportional to the direction cosines l,m,n
respectively are called DIRECTION RATIOS (d.r’s) of a line.
Note : If (a, b, c) are the direction ratios of a line then for any non-zero real
number λ , (λa, λb, λc) are also the direction ratios of the same line.
Direction cosines of a line in terms of its direction ratios
If (a, b, c) are direction ratios of a line then the direction cosines of the line are
a b c
± , ,
a +b +c a 2 + b2 + c 2 a 2 + b2 + c 2
2 2 2
THEOREM
The direction ratios of the line joining the points are ( x2 − x1 , y2 − y1 , z2 − z1 )
ANGLE BETWEEN TWO LINES
If (l1, m1, n1) and (l2, m2, n2) are the direction cosines of two lines θ and is the
acute angle between them, then cos θ = l1l2 + m1m2 + n1n2
Note.
If θ is the angle between two lines having d.c’s (l1, m1, n1) and (l2, m2, n2) then
sin θ = ∑ ( l1m2 − l2 m1 )2
and tan θ = ∑ (l m
1 2 - l2 m1 )2
when θ ≠
π
l1l2 + m1m2 + n1n2 2
Note 1 : The condition for the lines to be perpendicular is l1l2 + m1m2 + n1n2 = 0
l1 m1 n1
Note 2 : The condition for the lines to be parallel is = =
l2 m2 n2
THEOREM
If (a1, b1, c1) and (a2, b2, c2) are direction ratios of two lines and θ is the angle
a1a2 + b1b2 + c1c2
between them then cos θ =
a12 + b12 + c12 a22 + b22 + c22
Note 1 : If the two lines are perpendicular then a1a2 + b1b2 + c1c2 = 0
a1 b1 c1
Note 2 : If the two lines are parallel then = =
a2 b2 c2
Note 3 : If one of the angle between the two lines is θ then other angle is 1800 − θ
EXERCISE – 6(B)
I
1. Find the direction ratios of the line joining the points (3, 4, 0) are
(4, 4, 4)
2. The direction ratios of a line are ( −6, 2, 3) . Find the direction cosines.
Dividing with 36 + 4 + 9 = 7
6 2 3
Direction cosines of the line are − , ,
7 7 7
3. Find the cosine of the angle between the lines, whose direction cosines
are
1 1 1 1 1
, , and , , 0
3 3 3 2 2
1 1 1 1 1
Sol: D.cs of the given lines are , , and , , 0
3 3 3 2 2 .
1 1 1 1 1 1 2 4 2
cos θ = l1l2 + m1m2 + n1n2 = . + . + .0 = = = =
3 2 3 2 3 6 6 6 3
4. Find the angle between the lines whose direction ratios are
(1, 1, 2 ) ( 3, − 3, 0 )
Sol: D.rs of the given lines are (1, 1, 2 ) and ( 3, − 3, 0 )
Let θ be the angle between the lines. Then
cos θ =
a1a2 + b1b2 + c1c2
=
( )
1 3 + 1 − 3 + 2.0
=0
a +b +c
2
1
2
1
2
1 a +b +c
2
2
2
2
2
2
1 + 1+ 4 3 + 3
π
⇒ θ=
2
12 4 3 12 4 3 48 − 36 − 12
Now l1l2 + m1 m2 + n1 n2 = . − . − . = =0
13 13 13 13 13 13 169
d.rs of OP are 2, 3, 4
d.rs of OQ are 1, k, 1
⇒ 2 + 3 k + 4 = 0 ⇒ 3k = − 6 ⇒ k =−2
II.
1 If the direction ratios of a line are (3, 4, 0) find its direction cosines are
also the angles made the co-ordinate axes.
Dividing with 9 + 16 + 0 = 5
D.cs of the line are 3 , 4 , 0
5 5
If α , β , γ are the angles made by the line with the co-ordinate axes, then
3 4
cos α = cos β = cos γ = 0
5 5
3 4 π
α = cos −1 , β = cos −1 , γ =
5 5 2
2. Show that the line through the points (1, -1, 2) (3, 4, -2) is perpendicular
to the line through the points (0, 3, 2) and (3, 5, 6)
Sol:
Sol: Let l , m, n be the d.rs of the required line. This line is perpendicular to the
lines with d.rs (1, -1, 2) and (2, 1, -1)
∴ l − m + 2n = 0 and 2l + m − n = 0
l m n l m n
= = ⇒ = =
1− 2 4 +1 1+ 2 −1 5 3
Dividing with 1 + 25 + 9 = 35
1 5 3
d.cs of the required line are − , ,
35 35 35
5. Show that the points (2, 3, -4), (1, -2, 3) and (3, 8, -11) are collinear
6. Show that the points (4, 7, 8), (2, 3, 4), ( −1, − 2, 1) , (1, 2, 5 ) are vertices of a
parallelogram
AB = ( 4− 2 ) + ( 7− 3 ) + ( 8− 4 ) = 4 + 16 + 16= 36 = 6
2 2 2
BC = ( 2 + 1) + ( 3 + 2 ) + ( 4 − 1) = 9 + 25 +9 = 43
2 2 2
CD = ( −1, − 1) + ( −2 − 2 ) + (1 − 5 ) = 4 + 16 + 16 = 6
2 2 2
and
DA = (1 − 4 ) + ( 2− 7 ) + (5 − 8) = 9 + 25 + 9= 43
2 2 2
∴ AB = CD and BC = DA
III
1. Show that the lines whose direction cosines are given by l+m+n=0
2 m + 3nl − 5lm = 0 are perpendicular to each other
⇒ 2mn − 3n ( m + n ) + 5m ( m + n ) = 0
2
m m
⇒ 5 + 4 − 3 = 0
n n
m1m2 −3 mm nn
⇒ = ⇒ 1 2 = 1 2 − − − − − − ( 3)
n1n2 5 −3 5
From (1), n = − ( l + m)
⇒ 3l 2 + 10lm + 2m2 = 0
2
l l
⇒ 3 + 10 + 2 = 0
m m
l1l2 2 ll mm
⇒ = ⇒ 1 2 = 1 2 − − − − − ( 4)
m1m2 3 2 3
2.Find the angle between the lines whose direction cosines satisfy the equation
l + m + n = 0, l 2 + m 2 − n 2 = 0
l+m+n=0 ……….(1)
l 2 + m2− n2 = 0 ……..(2)
From (1), l = − ( m + n )
Substituting in (2)
( m + n ) + m2 − n2= 0
2
⇒ m2 + n2 + 2mn + m2 − n2 = 0
⇒ 2m 2 + 2mn = 0
⇒ 2m ( m + n )= 0
⇒ m = 0 and m + n = 0
l n
l=−n ⇒ =
1 −1
D.rs of the first line are (1, 0, − 1)
m n
Case (ii) : m + n = 0 ⇒ m = − n ⇒ =
1 −1
| 0 + 0 + 1| 1 π
= = ∴θ =
2. 2 2 3
3. If a ray makes angle α , β , γ and δ with the four diagonals of a cube find
cos2 α + cos 2 β + cos2 γ + cos2 δ
Sol:
Let a be the side of the cube. Let one of the vertices of the cube be the origin O
and the co-ordinate axes be along the three edges OA , OB and OC passing through
the origin.
The co-ordinate of the vertices of the cube with respect to the frame of reference
OABC are as shown in figure are A (a,o,o), B(o,a,o), C(0,o,a) P(a,a,a) Q(a,a,o)
R(o,a,a) and S(a,o,a)
The diagonals of the cube are OP , CQ , ARand BS . and their d.rs are respectively
Then l 2 + m 2 + n 2 = 1
If this ray is making the angles α , β , γ and δ with the four diagonals of the cube,
then
|a×l + a×m + a×n| |l + m + n|
cos α = =
a 2 + a 2 + a 2 .1 3
Similarly, cos β = | l + m − n |
3
| −l + m + n | | −l + m + n |
cos γ = and cos δ =
3 3
1
{| l + m + n |2 + | l + m − n |2 + | −l + m + n |2 + | l − m + n |2 }
3
1
[( l + m + n ) + ( l + m − n ) + ( −l + m + n ) + ( l − m + n ) ]
2 2 2 2
1
3
( )
[4 l 2 + m2 + n2 ] =
4
3
(since l 2 + m2 + n2 = 1)
4. If ( l1 , m1 , n1 ) , ( l2 , m2 , n2 ) and d.cs of two intersecting lines show that d.c.s of
two lines, bisecting the angles between them are proportional to
l1 ± l2 , m1 ± m2 n1 ± n2
Sol:
P1 = ( −l1 , − m1 , − n1 ) .
and
M = l1+ l2 , m1 + m2 , n1 + nl 2
2 2 2 ,
l − l m − m2 n1 − n2
M1 = 1 2 , 1 ,
2 2 2
5. A (-1, 2, -3), B(5, 0, -6), C(0, 4, -1) are three points. Show that the
direction cosines of the bisector of BAC are proportional to (25, 8, 5)
and (-11, 20, 23)
Sol: Given points are A(-1, 2, -3), B (5, 0, -6) and C(0, 4, -1)
6 1 −2 2 −3 2 18 + 7 −6 + 14 −9 + 14 25 8 5
= + , + , + = , , = , ,
7 3 7 3 7 3 21 21 21 21 21 21
6 1 −2 2 −3 2 18 − 7 −6 − 14 −23
l1− l2 , m1 − m2 , n1 − n2 = − , − , − = , ,
7 3 7 3 7 3 21 21 21
11 −20 −23
= , ,
21 21 21
Sol:
1.1 + 2 ( −2 ) + 3.1 π
cos ABC = =0 ⇒ B=
1+ 4 + 9 1+ 4 +1 2
7. The vertices of a triangle are A(1, 4, 2), B(-2, 1, 2) C(2, 3, -4). Find
A, B, C
Sol:
8. Find the angle between the lines whose direction cosines are given by
the equation 3l + m + 5n = 0 and 6 mn − 2 nl + 5l = 0
Sol: Given 3l + m + 5n = 0
6 mn − 2 nl + 5lm = 0
From (1), m = − ( 3l + 5n )
Substituting in (2)
⇒ −6n ( 3l + 5n ) − 2nl − 5l ( 3l + 5n ) = 0
⇒ l 2 + 3ln + 2n2 = 0
⇒ ( l + 2n ) ( l + n ) = 0
⇒ l + 2 n = 0 or l + n = 0
Case (i) :
l1 n1
l1+ n1 = 0 ⇒ n1= − l1; ⇒ n1 = − l1; ⇒ =
1 −1
m1 n1
∴ =
+2 −1
l1 m1 n1
∴ = =
1 2 −1
D.rs of the first line l1 are (1, 2, − 1)
l2 n2
⇒ l2 = − 2n2 ⇒ =
−2 1
m2 n2
=
1 1
l2 m2 n2
∴ = =
−2 1 1
Sol: The d.cs of the line in the two positions are ( l , m, n ) and ( l + δ l , m + δ n, n + n )
.
and ( l + δ l 2 ) + ( m + δ m ) + ( n + δ n ) = 1 ----(2)
2 2
( 2 ) − (1) ⇒ ( l+ δ l ) + ( m + δ m ) + ( n + δ n ) − ( l 2 + m2 + n2 ) = 0
2 2 2
( )
= l 2 + m 2 + n 2 + ( l .δ l + m.δ m + n.δ n )
1
cos δθ = 1 − (δ l ) + (δ m ) + (δ n )
2 2 2
2
(δ l ) + (δ m ) + (δ n ) = 2 (1 − cos δθ )
2 2 2
δθ
= 2. 2sin 2
2
δθ δθ
δθ being small, sin =
2 2
δθ
2
∴ 4 sin θ = 4 = (δθ )
2 2
2
∴ (δθ ) = (δ l ) + (δ m ) + (δ n )
2 2 2 2
6. A(1,8,4), B(0,–11,4), C(2,–3,1) are three points and D is the foot of the
perpendicular from A to BC. Find the coordinates of D.
Solution: -
suppose D divides in the ratio m : n
æ 2m ö
- 3m - 11n m + 4n ÷
Then D = ççç , , ÷
èm + n m+ n m+ n ÷ø
æm - n - 11m - 19n - 3m ÷ö
Direction ratios of AD = ççç , , ÷
èm + n m+ n m + n÷
ø
Direction ratios of BC : (2,8, - 3)
æm - n÷ö æ- 11m - 19n ö÷ æ- 3m ö÷
AD ^ BC Þ 2 çç ÷+ 8çç ÷- 3çç ÷= 0
çè m + n ø èç m + n ø÷ èç m + n ø÷
÷
2m–2n–88m–152n+9m=0
m = –2n
substituting in (1), D=(4,5,–2)
7. Lines OA, OB are drawn from O with direction cosines proportional to (1,–2,–1);
(3,–2,3). Find the direction cosines of th enormal to the plane AOB.
Sol : -
Let (a, b, c) be the direction ratios of a normal to the plane AOB. since
OA, OB lie on the plane, they are perpendicular to the normal to the plane.
Using the condition of perpendicularity
a.1+b(–2)+ c (–1) = 0. ...... (1)
a.3 + b(–2) +c(3) = 0. ...... (2)
a b c a b c
Solving (1) and (2) = = or = =
- 8 - 6 4 4 3 - 2
The d.c’s of the normal are
æ 4 3 - 2 ö
÷ æ 4 3 - 2 ö÷
çç , , ÷i.e., çç , , ÷
çè 16 + 9 + 4 16 + 9 + 4 16 + 9 + ÷ çè 29 29 29 ø÷
4÷
ø
8. Show that the line whose d.c’s are proportional to (2,1,1), (4, 3 - 1, - 3 - 1) are
inclined to one another at angle .
9. Find the d.r’s and d.c’s of the line joining the points (4, –7, 3), (6, –5, 2)
10. For what value of x the line joining A(4,1,2) B(5,x,0) is perpendicular to the
line joining C(1,2,3) and D(3,5,7).
11. Find the direction cosines of two lines which are connected by the relations
l + m + n = 0 and mn-2nl-2lm = 0