Chapter 10
VECTOR ALGEBRA
10.1 Overview
10.1.1 A quantity that has magnitude as well as direction is called a vector.
a
10.1.2 The unit vector in the direction of a is given by | a | and is represented by a .
10.1.3 Position vector of a point P (x, y, z) is given as OP = xiˆ + y ˆj+ z kˆ and its
magnitude as | OP | = x 2 + y 2 + z 2 , 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.
, VECTOR ALGEBRA 205
10.1.9 Vector joining two points
If P1 (x1, y1,z1) and P2 (x2, y2,z2) are any two points, then
P1P2 = ( x2 − x1 ) iˆ + ( y2 − y1 ) ˆj + ( z2 − z1 ) kˆ
| P1P2 | = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2
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
na + mb
(i) in the ratio m : n internally, is given by
m+n
mb – na
(ii) in the ratio m : n externally, is given by
m–n
a. b
10.1.11 Projection of a along b is and the Projection vector of a along b
|b|
a . b
is 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 n̂ ,
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where n̂ is a unit vector perpendicular to the plane containing a and b and a , b , n̂
form a right handed system.
10.1.14 If a = a1 iˆ + a2 ˆj + a3 kˆ and b = b1 iˆ + b2 ˆj + b3 kˆ are two vectors and λ is
any scalar, then
a + b = (a1 + b1 ) iˆ + (a2 + b2 ) ˆj + (a3 + b3 ) kˆ
λ a = (λ a1 ) iˆ + (λ a2 ) ˆj + (λ a3 ) kˆ
a . b = a1 b1+ a2 b2 + a3 b3
iˆ ˆj kˆ
a b1 c1
a ×b = 1 = (b1c2 – b2c1) iˆ + (a2c1 – c1c2) ĵ + (a1bb – a2b1) k̂
a2 b2 c2
Angle between two vectors a and b is given by
a1 b1 + a2 b2 + a3b3
a. b
cos θ = | | | | = 2
a b a1 + a22 + a32 b12 + b22 + b32
10.2 Solved Examples
Short Answer (S.A.)
Example 1 Find the unit vector in the direction of the sum of the vectors
a = 2 iˆ − ˆj + 2 kˆ and b = – iˆ + ˆj + 3 kˆ .
Solution Let c denote the sum of a and b . We have
c = (2 iˆ − ˆj + 2 kˆ) + (−iˆ + ˆj + 3 kˆ) = iˆ + 5 kˆ
Now | c | = 12 + 52 = 26 .
VECTOR ALGEBRA
10.1 Overview
10.1.1 A quantity that has magnitude as well as direction is called a vector.
a
10.1.2 The unit vector in the direction of a is given by | a | and is represented by a .
10.1.3 Position vector of a point P (x, y, z) is given as OP = xiˆ + y ˆj+ z kˆ and its
magnitude as | OP | = x 2 + y 2 + z 2 , 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.
, VECTOR ALGEBRA 205
10.1.9 Vector joining two points
If P1 (x1, y1,z1) and P2 (x2, y2,z2) are any two points, then
P1P2 = ( x2 − x1 ) iˆ + ( y2 − y1 ) ˆj + ( z2 − z1 ) kˆ
| P1P2 | = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2
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
na + mb
(i) in the ratio m : n internally, is given by
m+n
mb – na
(ii) in the ratio m : n externally, is given by
m–n
a. b
10.1.11 Projection of a along b is and the Projection vector of a along b
|b|
a . b
is 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 n̂ ,
, 206 MATHEMATICS
where n̂ is a unit vector perpendicular to the plane containing a and b and a , b , n̂
form a right handed system.
10.1.14 If a = a1 iˆ + a2 ˆj + a3 kˆ and b = b1 iˆ + b2 ˆj + b3 kˆ are two vectors and λ is
any scalar, then
a + b = (a1 + b1 ) iˆ + (a2 + b2 ) ˆj + (a3 + b3 ) kˆ
λ a = (λ a1 ) iˆ + (λ a2 ) ˆj + (λ a3 ) kˆ
a . b = a1 b1+ a2 b2 + a3 b3
iˆ ˆj kˆ
a b1 c1
a ×b = 1 = (b1c2 – b2c1) iˆ + (a2c1 – c1c2) ĵ + (a1bb – a2b1) k̂
a2 b2 c2
Angle between two vectors a and b is given by
a1 b1 + a2 b2 + a3b3
a. b
cos θ = | | | | = 2
a b a1 + a22 + a32 b12 + b22 + b32
10.2 Solved Examples
Short Answer (S.A.)
Example 1 Find the unit vector in the direction of the sum of the vectors
a = 2 iˆ − ˆj + 2 kˆ and b = – iˆ + ˆj + 3 kˆ .
Solution Let c denote the sum of a and b . We have
c = (2 iˆ − ˆj + 2 kˆ) + (−iˆ + ˆj + 3 kˆ) = iˆ + 5 kˆ
Now | c | = 12 + 52 = 26 .
