CH APTER
VECTORS
Vector Quantities
A physical quantity which requires magnitude and a
particular direction, when it is expressed.
Triangle law of Vector addition
r r r
R = A+B
R=
2
If some vectors are represented by sides of a polygon in
same order, then their resultant vector is represented by
the closing side of polygon in the opposite order.
C
D
2
A + B + 2AB cos q
tan a =
Addition of More than Two Vectors
(Law of Polygon)
R
B sin q
A + B cos q
q
a
A
C
B Bsinq
B
A
Bcosq
q
q
& a=
2
2
Rmax = A+B for q=0° ;Rmin = A–B for q=180°
D
r r r r r
R = A+B+C+D
If A = B then R = 2A cos
Parallelogram Law of Addition
of Two Vectors
r
R
b
A
q
B
A
r
B sin q
A sin q
and tan b =
A + B cos q
B + A cos q
Vector subtraction
Bcosq
a
Bsin q
R
q
= m
A 2x + A 2y + A 2z
Az
2
2
2
2
x
= n
A 2x + A 2y + A 2z
( A +A +A )
2
x
2
y
2
z
2
=1
or sin2a + sin2b + sin2g =2
y
General Vector in x-y plane
r
r = xiˆ + yjˆ = r cos qˆi + sin qˆj
(
x
)
EXAMPLES :
1. Construct a vector of magnitude 6 units making
an angle of 60° with x-axis.
B
B sin q
A2 + B2 - 2AB cos q , tan a = A - B cos q
If A = B then R = 2A sin
2
q
q
r r r
r r
r
R = A - B Þ R = A + ( -B )
R=
2
r
B
= l
A + A2y + A 2z
A + A 2y + A 2z
l, m, n are called direction cosines
l +m +n =cos a+cos b+cos g=
2
Ax
2
x
Ay
Angle made with z-axis
A
cos g = z =
A
uuur uuur uuur ur
r r r
AB + AD = AC = R or A + B = R Þ R = A 2 + B 2 + 2AB cos q
tan a =
Angle made with y-axis
Ay
cos b =
=
A
+B
=A
a
r
A = A xˆi + A y ˆj + A z kˆ
Angle made with x-axis
A
cos a = x =
A
C
B
B
A
Rectangular component of a 3–D vector
If two vectors are represented by two adjacent sides of a
parallelogram which are directed away from their common
point then their sum (i.e. resultant vector) is given by the
diagonal of t he parallelogram passing away through that
common point.
D
R
q
2
r
ˆ = 6 æ 1 ˆi + 3 ˆj ö = 3iˆ + 3 3ˆj
Sol. r = r(cos 60iˆ + sin 60j)
çè
÷ø
2
2
2. Construct an unit vector making an angle of 135°
with x axis.
1 ˆ ˆ
Sol. r̂ = 1(cos135°ˆi + sin135°ˆj) =
( - i + j)
2
Differentiation
Scalar product (Dot Product)
r
r
r r
r r
Angle between
æ
ö
-1 A × B
A.B = AB cos q Þ
q = cos ç
è AB ÷ø
two vectors
r
r
r
If A = A xˆi + A y ˆj + A z kˆ & B = B xˆi + B yˆj + B zkˆ then
r
r r
A.B = A x Bx + A y B y + A z Bz and angle between
r
r
A & B is given by
r r
A.B
cos q =
=
AB
r
r
When a particle moved from
(x1, y1, z1) to (x2, y2, z2) then its
displacement vector
A x Bx + A y B y + A z Bz
(x1,y1,z1) r
A2x + A 2y + A 2z B2x + B2y + B2z
ˆ ˆ = 1 , ˆi.jˆ = 0 , î.kˆ = 0 , ĵ.kˆ = 0
ˆi.iˆ = 1 , ˆj.jˆ = 1 , k.k
r
r
Component of vector b along vector a ,
r
r
b||= b . aˆ aˆ
(
)
r r r
ˆ - (x ˆi + y ˆj + z k)
ˆ
r = r2 - r1 = (x 2ˆi + y 2ˆj + z 2 k)
1
1
1
Lami's theorem
a
A
r
r
Component of b perpendicular to a ,
c
Cross Product (Vector product)
r r
ˆ where n̂ is a vect or
A ´ B = AB sin q n
r
r
perpendicular to A & B or their plane and its
direction given by right hand thumb rule.
B
B
C
r
Bx
By
Bz
B
q
r
ˆi ´ ˆj = kˆ ; ˆj ´ kˆ = ˆi ,
i
negative
kˆ ´ ˆi = ˆj ; ˆj ´ ˆi = -kˆ
kˆ ´ ˆj = -ˆi , ˆi ´ kˆ = -ˆj
F3
k
B
Bsinq
q
A
Area of parallelogram
r
r
q1
F1
F
F3
= 2 =
sin q1 sin q2 sin q 3
sin A sin B sin C
=
=
a
b
c
B
= ˆi ( A y B z - A z B y ) - ˆj (AxBz–BxAz) + k̂ (AxBy–BxAy)
r r
r r
A ´ B = -B ´ A
r r r
r r r
j
(A ´ B).A = (A ´ B).B = 0
r
r
r
ˆi ´ ˆi = 0 , ˆj ´ ˆj = 0 , kˆ ´ kˆ = 0
positive
r
C
a
r r
A´B 1
Area =
= AB sin q
2
2
B
kˆ
Az
q2
Area of triangle
A
ˆj
Ay
q3
b
A ×B
ˆi
r r
A ´ B = Ax
F2
F1
A
r
r r
r
r
b ^ = b - b|| = b - ( b × aˆ ) aˆ
A
r2
r
Magnitude: r = r = (x2 - x1 )2 + (y2 - y1 )2 + (z2 - z1 )2
b
r
(x2,y2,z2)
r1
= (x2 - x1 )iˆ + (y 2 - y1 )ˆj + (z 2 - z1 )kˆ
b
r
r
r
d r r
dA r r dB
(A.B) =
.B + A.
dt
dt
dt
r
r
d r r
dA r r dB
(A ´ B) =
´B+A´
dt
dt
dt
Bsinq
r r
Area = A ´ B = ABsinq
A
For parallel vectors
r r r
A´B = 0
For perpendicular vectors
r r
A.B = 0
For coplanar vectors
r r r
A.(B ´ C) = 0
If A,B,C points are collinear
uuur
uuur
AB = l BC
Examples
of
dot
products
w
rr
Work, W = F.d = Fdcosq
where
F ® force, d ® displacement
w
rr
Power, P = F.v = Fvcosq
where
F ® force, v ® velocity
w
r r
Electric flux, fE = E.A = EAcosq
where
E ® electric field, A ® Area
w
r r
Magnetic flux, fB = B.A = BAcosq
where
B ® magnetic field, A ® Area
w
Potential energy of dipole in
where
p ® dipole moment,
r r
uniform field, U = – p.E
where
E ® Electric field
w
r
Torque rt = rr ´ F
w
r r r
Angular momentum L = r ´ p where r ® position vector, p ® linear momentum
w
r
Linear velocity vr = w ´ rr where r ® position vector, w ® angular velocity
w
r r r
Torque on dipole placed in electric field t = p ´ E
where r ® position vector, F ® force
Examples
of
cross
products
where p ® dipole moment, E ® electric field
KEY
S
INT
O
P
•
Tensor : A quantity that has different values in different directions is called tensor.
Example : Moment of Inertia
In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and
a vector is a first rank tensor.
•
Electric current is not a vector as it does not obey the law of vector addition.
•
A unit vector has no unit.
•
To a vector only a vector of same type can be added and the resultant is a vector of the same type.
•
A scalar or a vector can never be divided by a vector.