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10.1 Overview
10.1.1 A quantity that has magnitude as well as direction is called a vector.
10.1.2 The unit vector in the direction of a
is given by | |
a
a
and is represented by a
10.1.3 Position vector of a point P ( x , y , z ) is given as
OP = xi + y j + z k
and its
magnitude as
2 2 2
| OP | = x + y + z
, where O is the origin.
10.1.4 The scalar components of a vector are its direction ratios, and represent its
projections along the respective axes.
10.1.5 The magnitude r , direction ratios ( a , b , c ) and direction cosines ( l , m , n ) of any
vector are related as:
a b c
l m n
r r r
10.1.6 The sum of the vectors representing the three sides of a triangle taken in order is 0
10.1.7 The triangle law of vector addition states that “If two vectors are represented
by two sides of a triangle taken in order, then their sum or resultant is given by the third
side taken in opposite order”.
10.1.8 Scalar multiplication
If a
is a given vector and λ a scalar, then λ a
is a vector whose magnitude is |λ a
| = |λ|
| a
|. The direction of λ a
is same as that of a
if λ is positive and, opposite to that of a
if
λ is negative.
Chapter 10
VECTOR ALGEBRA
VECTOR ALGEBRA 205
10.1.9 Vector joining two points
If P 1
( x 1
, y 1
, z 1
) and P 2
( x 2
, y 2
, z 2
) are any two points, then
1 2 2 1 2 1 2 1
P P = ( x − x ) i + ( y − y ) j + ( z − z ) k
2 2 2
1 2 2 1 2 1 2 1
| P P | = ( x − x ) + ( y − y ) + ( z − z )
10.1.10 Section formula
The position vector of a point R dividing the line segment joining the points P and Q
whose position vectors are (^) a
and b
(i) in the ratio m : n internally, is given by
na mb
m n
(ii) in the ratio m : n externally, is given by
mb na
m n
10.1.11 Projection of a
along (^) b
is
a (^) b
b
^
(^) and the Projection vector of a
along (^) b
is
a (^) b
b
^
b
10.1.12 Scalar or dot product
The scalar or dot product of two given vectors a
and (^) b
having an angle θ between
them is defined as
a
. (^) b
= | a
| | (^) b
| cos θ
10.1.13 Vector or cross product
The cross product of two vectors a
and (^) b
having angle θ between them is given as
a
× (^) b
= | a
| | (^) b
| sin θ n ˆ ,
VECTOR ALGEBRA 207
Thus, the required unit vector is
( )
c
c i k i k
c
Example 2 Find a vector of magnitude 11 in the direction opposite to that of PQ
where P and Q are the points (1, 3, 2) and (–1, 0, 8), respetively.
Solution The vector with initial point P (1, 3, 2) and terminal point Q (–1, 0, 8) is given by
PQ
i
j + (8 – 2)^
k
i
j + 6^
k
Thus Q P
– P Q
2 i + 3 j − 6 k
2 2 2
| | 2 3 (–6) 4 9 36 49 7 QP
Therefore, unit vector in the direction of (^) Q P
is given by
Q P
Q P
| |^7
Q P
i + j − k
Hence, the required vector of magnitude 11 in direction of (^) Q P
is
QP
ˆ ˆ ˆ 2 3 6
7
(^) i + j − k
i + j k.
Example 3 Find the position vector of a point R which divides the line joining the two
points P and Q with position vectors (^) O P =^2 a^ + b
and (^) O Q = a^ – 2 b
, respectively,,
in the ratio 1:2, (i) internally and (ii) externally.
Solution (i) The position vector of the point R dividing the join of P and Q internally in
the ratio 1:2 is given by
O R
a a a b b
208 MATHEMATICS
(ii) The position vector of the point R′ dividing the join of P and Q in the ratio
1 : 2 externally is given by
O R
a a b b
a b
Example 4 If the points (–1, –1, 2), (2, m , 5) and (3,11, 6) are collinear, find the value of m.
Solution Let the given points be A (–1, –1, 2), B (2, m , 5) and C (3, 11, 6). Then
ˆ ˆ^ ˆ
A B = (2 + 1) i + ( m + 1) j +(5 – 2) k
3 i + ( m + 1) j + 3 k
and
A C = (3 + 1) i + (11 + 1) j + (6 −2) k
4 i + 12 j + 4 k.
Since A, B, C, are collinear, we have (^) A B
= λ A C
, i.e.,
ˆ ˆ ˆ^ ˆ ˆ ˆ
3 i + ( m + 1) j + 3 ) k =λ (4 +12 + 4 ) i j k
⇒ 3 = 4 λ and m + 1 = 12 λ
Therefore m = 8.
Example 5 Find a vector r
of magnitude 3 2
units which makes an angle of
π
and
π
with y and z - axes, respectively..
Solution Here m =
π 1
cos
and n = cos
π
2
Therefore, l
2
2
2 = 1 gives
l
2
1
2
⇒ l = ±
210 MATHEMATICS
Therefore, unit vector perpendicular to the plane of (^) a
and (^) b
is given by
ˆ ˆ^ ˆ
a b i j k
a b
× − +
×
Hence, vectors of magnitude of
that are perpendicular to plane of a
and (^) b
are
ˆ ˆ^ ˆ
i j k
, i.e.,
± 10( i − j + k ).
Long Answer (L.A.)
Example 8 Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.
Solution Let
OP and^
OQ be unit vectors making angles A and B, respectively, with
positive direction of x -axis. Then ∠QOP = A – B [Fig. 10.1]
We know
OP =^
OM + MP = i cos A + j sin A
and
OQ =^
ON + NQ = i cos B + j sin B.
By definition
^
OP. OQ =OP OQ cos ( A-B)
= cos (A – B) ... (1)
( )
OP = = 1 OQ
In terms of components, we have
OP. OQ =^
( cos A i + j sin A).( cos B i + j sin B)
= cosA cosB + sinA sinB ... (2)
From (1) and (2), we get
cos (A – B) = cosA cosB + sinA sinB.
VECTOR ALGEBRA 211
Example 9 Prove that in a ∆ ABC,
sin A sin B sin C
a b c
, where a , b , c represent the
magnitudes of the sides opposite to vertices A, B, C, respectively.
Solution Let the three sides of the triangle BC, CA and AB be represented by
a b , and c
, respectively [Fig. 10.2].
We have (^) a + b + c = 0
^ ^
. i.e., (^) a + b = − c
which pre cross multiplying by (^) a
, and
post cross multiplying by b
, gives
a × b = c × a
and (^) a × b = b × c
respectively. Therefore,
a × b = b × c = c × a
⇒ a × b = b × c = c × a
⇒ a b sin ( π – C) = b c sin ( π – A) = c a sin ( π– B)
⇒ ab sin C = bc sinA = ca sinB
Dividing by abc , we get
sin C sin A sin B
c a b
= = (^) i.e.
sin A sin B sin C
a b c
Objective Type Questions
Choose the correct answer from the given four options in each of the Examples 10 to 21.
Example 10 The magnitude of the vector
6 i + 2 j + 3 k is
VECTOR ALGEBRA 213
Example 15 The area of the parallelogram whose adjacent sides are
i + k and
ˆ ˆ^ ˆ
2 i + j + k is
(A)
(B) 3 (C) 3 (D) 4
Solution (B) is the correct answer. Area of the parallelogram whose adjacent sides
are (^) a and b
is
a × b
Example 16 If a
= 8, b = 3
and a × b = 12
, then value of (^) a b.
is
(A)
(B)
(C)
(D) None of these
Solution (C) is the correct answer. Using the formula a × b = a. b
|sinθ|, we get
π
θ= ± (^).
Therefore, (^) a b.
= a^.^ b^ cosθ
= 8 × 3 ×
Example 17 The 2 vectors
ˆ ˆ^ ˆ ˆ ˆ
j + k and 3 i − j + 4 k represents the two sides AB and
AC, respectively of a ∆ABC. The length of the median through A is
(A)
(B)
(C)
(D) None of these
Solution (A) is the correct answer. Median AD
is given by
AD 3 5
= i + j + k =
Example 18 The projection of vector
a = 2 i − j + k
along
b = i + 2 j + 2 k
is
214 MATHEMATICS
(A)
(B)
(C) 2 (D) 6
Solution (A) is the correct answer. Projection of a vector (^) a on b is
a b.
b
i − j + k i + j + k
Example 19 If (^) a and b
are unit vectors, then what is the angle between (^) a and b
for
3 a − b
to be a unit vector?
(A) 30° (B) 45° (C) 60° (D) 90°
Solution (A) is the correct answer. We have
2 2 2
( 3 a − b ) = 3 a + b − 2 3 a b.
⇒ (^) a b.
⇒ cosθ =
⇒ θ = 30°.
Example 20 The unit vector perpendicular to the vectors
i − j and^
i + j forming a
right handed system is
(A) ˆ
k
(B) – ˆ
k
(C)
i − j
(D)
i + j
Solution (A) is the correct answer. Required unit vector is
( ) ( )
( ) ( )
i j i j
i j i j
− × +
− × +
k
= k.
Example 21 If a = 3
and (^) – 1 ≤ k ≤ 2 , then ka
lies in the interval
(A) [0, 6] (B) [– 3, 6] (C) [ 3, 6] (D) [1, 2]
216 MATHEMATICS
12. If A, B, C, D are the points with position vectors
i + j − k ,^
2 i − j + 3 k ,
ˆ ˆ^ ˆ ˆ ˆ
2 i − 3 , 3 k i − 2 j + k , respectively, find the projection of^ AB
along (^) CD
13. Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3),
B(2, – 1, 4) and C(4, 5, – 1).
14. Using vectors, prove that the parallelogram on the same base and between the
same parallels are equal in area.
Long Answer (L.A.)
15. Prove that in any triangle ABC,
2 2 2
cos A
b c a
bc
= (^) , where a , b , c are the
magnitudes of the sides opposite to the vertices A, B, C, respectively.
16. If (^) a b c , ,
determine the vertices of a triangle, show that
b (^) × c + c × a + a × b
gives the vector area of the triangle. Hence deduce the
condition that the three points (^) a b c , ,
are collinear. Also find the unit vector normal
to the plane of the triangle.
17. Show that area of the parallelogram whose diagonals are given by (^) a
and b
is
a × b
. Also find the area of the parallelogram whose diagonals are
2 i − j + k
and
i + 3 j − k.
18. If a
i + j + k and^
b = j − k
, find a vector c
such that (^) a × c = b
and a c. = 3
Objective Type Questions
Choose the correct answer from the given four options in each of the Exercises from
19 to 33 (M.C.Q)
19. The vector in the direction of the vector
i − 2 j + 2 k that has magnitude 9 is
(A)
i − 2 j + 2 k (B)^
i − j + k
(C)
3( i − 2 j + 2 ) k (D)^
9( i − 2 j +2 ) k
VECTOR ALGEBRA 217
20. The position vector of the point which divides the join of points (^2) a − 3 b
and (^) a + b
in the ratio 3 : 1 is
(A)
a − b
(B)
a − b
(C)
a
(D)
a
21. The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively
is
(A)
− + i 12 j + 4 k (B)^
5 i + 2 j − 4 k
(C)
− 5 i + 2 j + 4 k (D)^
i + j + k
22. The angle between two vectors (^) a and b
with magnitudes 3
and 4, respectively,,
and (^) a b. = 2 3
is
(A)
π
(B)
π
(C)
π
(D)
π
23. Find the value of λ such that the vectors
a = 2 i + λ + j k
and
b = i + 2 j + 3 k
are
orthogonal
(A) 0 (B) 1 (C)
(D) –
24. The value of λ for which the vectors
ˆ ˆ ˆ^ ˆ ˆ ˆ
3 i − 6 j + k and 2 i − 4 j + λ k are parallel is
(A)
(B)
(C)
(D)
25. The vectors from origin to the points A and B are
ˆ ˆ ˆ^ ˆ ˆ ˆ
a = 2 i − 3 j + 2 k and b = 2 i + 3 j + k
,respectively, then the area of triangle OAB is
(A) 340 (B)
(C)
(D)
VECTOR ALGEBRA 219
35. If (^) r a. = 0, r b. = 0, and r c. = 0
for some non-zero vector r
, then the value of
a .( b × c )
is ________
36. The vectors
a = 3 i − 2 j + 2 k
and
b = – i − 2 k
are the adjacent sides of a
parallelogram. The acute angle between its diagonals is ________.
37. The values of k for which
and
ka < a ka + a
is parallel to (^) a
holds true
are _______.
38. The value of the expression
2 2
a × b + ( a b. )
is _______.
39. If
2 2
a × b + a b.
= 144 and a = 4
, then b
is equal to _______.
40. If a
is any non-zero vector, then ( ) (^) ( )
ˆ ˆ ˆ ˆ ˆ^ ˆ
( a i i. ) + a j. j + a k k.
equals _______.
State True or False in each of the following Exercises.
41. If a^ =^ b
, then necessarily it implies (^) a = ± b
42. Position vector of a point P is a vector whose initial point is origin. 43. If a + b = a − b
, then the vectors a
and b
are orthogonal.
44. The formula
2 2 2
( a + b ) = a + b + 2 a × b
is valid for non-zero vectors (^) a
and b
45. If (^) a
and b
are adjacent sides of a rhombus, then (^) a
b
