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Scalars and Vectors
Scalars are physical quantities specified completely by magnitude
i.e. a number possibly followed by a unit of measurement.
Examples :
Distance to brightest star Sirius – 8.6 ly
Average rainfall in Cherrapunji – 11777 mm
Hottest spot on earth (Lut Desert, Iran) – 70
o C
Heaviest weight lifted (Clean & jerk by Rezazadeh, Iran) – 263 kg
Maximum speed in animal kingdom (Peregrine falcon) – 389 km/h
Physics examples – work done, flux, mass, electrical resistance etc.
Properties :
I (^) Remain unchanged under coordinate transformation.
I Independent of coordinate system used.
I (^) Ordinary rules of arithmetic apply – addition & subtraction,
multiplication & division.
Scalars and Vectors
Vectors are physical quantities specified completely by magnitude
and direction w.r.t. some reference.
a
^
|A|
A −A
|A|
−a
^
Examples :
Displacement, Area, Velocity, Momentum, Force, Torque, Electric
field, Magnetic field, Runge-Lenz vector etc.
Properties :
I (^) May change direction under coordinate transformation
(rotation).
I (^) Denoted by
A or A whose magnitude is |
A| or A and direction
given by
A or more often by unit vector ˆa =
A/|
A|.
I (^) Represented by a directed line i.e. an arrow in direction of ˆa.
Vectors – additional properties
Result of vector addition (for dot product wait till next slide):
C =
A +
B and C =
C ·
C
C
2 = A
2
2
B
C
sin θ
Sometimes (or more familiar), angle φ expressed in terms
A and
B
tan φ =
B sin θ
A + B cos θ
=
A
B
A A
B B
C C
(A+B)+C
A+(B+C)
C=A+B
A+B
B+C
=
C=A+B
B
A
- Commutative:
A +
B =
B +
A
- Associative: (
A +
B) +
C =
A + (
B +
C )
A few more rules of vector algebra...
Vector multiplication –
- Scalar or Dot product:
A ·
A = |
A||
B| cos θ
1.1 Results in a scalar quantity.
1.2 Gives projection of one vector on another.
~ A ·
~ B = 0 and
~ A 6 = 0 6 =
~ B ⇒ θ = π/2 meaning
~ A ⊥
~ B.
1.4 Commutative:
~ A ·
~ B =
~ B ·
~ A
1.5 Distributive:
~ A · (
~ B +
~ C ) =
~ A ·
~ B +
~ A ·
~ C
1.6 Examples: Mechanical work =
~ F · ~s, Magnetic flux =
~ B · ~a.
- Vector or Cross product:
A ×
B = ˆn|
A||
B| sin θ
2.1 Results in a vector quantity.
2.2 Gives area of a parallelogram of sides A and B.
~ A ×
~ B = 0 and
~ A 6 = 0 6 =
~ B ⇒ θ = nπ meaning
~ A ‖
~ B.
2.4 Not commutative:
~ A ×
~ B = −
~ B ×
~ A
2.5 Distributive:
~ A × (
~ B +
~ C ) =
~ A ×
~ B +
~ A ×
~ C
2.6 Examples: Angular momentum = ~r ×~p, Lorentz force = ~v ×
~ B.
Vector applications: Example 2.
Velocity & acceleration in uniform circular motion
r
r’
v
v’
v’ (^) v
R
∆ v
∆ r
P
P’
An object in uniform circular motion, at two instants of time, is at
P and P
′ of position vectors ~r and ~r
′ ( |~r | = |~r
′ | = R) having
velocities ~v and ~v
′ (|~v | = |~v
′ | = v ).
Triangle law of vector addition : ~v
′ − ~v = ∆~v & ~r
′ − ~r = ∆~r
∆~v ⊥ ∆~r and if ∆t is lapsed from P to P
′ →
average acceleration ~a av
= ∆~v /∆t i.e. |~a av
| = |∆~v |/∆t.
Similar triangles : |∆~v |/v = |∆~r |/R →
|~aav| =
v
R
|∆~r |
∆t
⇒ |~a| =
v
R
lim
∆t→ 0
|∆~r |
∆t
=
v
2
R
Vectors : components and bases
Practical necessities –
- knowing direction in conventional sense
- figuring resultant of complicated vector system
Bring in coordinate system – Cartesian system for instance
Components of vector
A – projections of
A along positive
directions of the coordinate axes (x, y , z) → (A x
, A
y
, A
z
I
A = (A
x
, A
y
, A
z
), A =
A
2
x
+ A
2
y
+ A
2
z
I A
x
= A cos α, A y
= A cos β, A z
= cos γ, where α, β, γ are
direction cosines.
I
A =
B ⇒ A
x
= B
x
, A
y
= B
y
, A
z
= B
z
I
A ·
B = A
x
B
x
+ A
y
B
y
+ A
z
B
z
m=x,y ,z
A
m
B
m
I
A ×
B = wait a while!!
Something more about vectors...
Vectors as row / column matrices – Consider two vectors
A = (Ax , Ay , Az ) and
B = (Bx , By , Bz ) and their scalar product
can be written as,
~ A =
Ax
Ay
Az
,
~ B =
Bx
By
Bz
⇒
~ A ·
~ B = (Ax Ay Az )
Bx
By
Bz
(^) = A x Bx +^ Ay By +^ Az Bz
Can you tell how unit vectors would be in column matrix notation?
Rotation of vector – Rotating a vector in a fixed basis is called
active transformation. An example is rotation of vector about
z-axis – active transformation through angle θ (counterclockwise),
A
′
x =^ Ax cos^ θ^ −^ Ay sin^ θ
A
′
y =^ Ax sin^ θ^ +^ Ay cos^ θ
A
′
z
= Az
⇒
A
′
x
A
′
y
A
′
z
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
Ax
Ay
Az
.
Vector applications: Examples
An object, starting from origin (0,0), made the following
displacements ~s 1
= (4, 4), ~s 2
= (6, −2), ~s 3
= (9, 8). Find how far
the object is from the origin and at what direction (w.r.t. x-axis)?
Vector applications: Examples
An object, starting from origin (0,0), made the following
displacements ~s 1
= (4, 4), ~s 2
= (6, −2), ~s 3
= (9, 8). Find how far
the object is from the origin and at what direction (w.r.t. x-axis)?
Net displacement
~s = ~s 1
= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10
Distance from origin |~s| =
2
2 = 21.5 units.
Vector applications: Examples
An object, starting from origin (0,0), made the following
displacements ~s 1
= (4, 4), ~s 2
= (6, −2), ~s 3
= (9, 8). Find how far
the object is from the origin and at what direction (w.r.t. x-axis)?
Net displacement
~s = ~s 1
= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10
Distance from origin |~s| =
2
2 = 21.5 units.
Direction i.e. at an angle w.r.t. x-axis
cos θ = (~s ·
i)/(|~s||
i|) = 0. 885 ⇒ θ = 27. 75
o
Vector applications: Examples
An object, starting from origin (0,0), made the following
displacements ~s 1
= (4, 4), ~s 2
= (6, −2), ~s 3
= (9, 8). Find how far
the object is from the origin and at what direction (w.r.t. x-axis)?
Net displacement
~s = ~s 1
= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10
Distance from origin |~s| =
2
2 = 21.5 units.
Direction i.e. at an angle w.r.t. x-axis
cos θ = (~s ·
i)/(|~s||
i|) = 0. 885 ⇒ θ = 27. 75
o
Two particles A and B move in opposite directions along a circle,
of radius R, with constant angular speed ω. At t = 0, they are
both at point ~r = R
j. Find the velocity of A w.r.t. B.
Relative separation
d = 2R sin(ωt)ˆi ⇒ ~v A/B
d = 2Rω cos(ωt)ˆi
Vector applications: Examples
An object, starting from origin (0,0), made the following
displacements ~s 1
= (4, 4), ~s 2
= (6, −2), ~s 3
= (9, 8). Find how far
the object is from the origin and at what direction (w.r.t. x-axis)?
Net displacement
~s = ~s 1
= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10
Distance from origin |~s| =
2
2 = 21.5 units.
Direction i.e. at an angle w.r.t. x-axis
cos θ = (~s ·
i)/(|~s||
i|) = 0. 885 ⇒ θ = 27. 75
o
Two particles A and B move in opposite directions along a circle,
of radius R, with constant angular speed ω. At t = 0, they are
both at point ~r = R
j. Find the velocity of A w.r.t. B.
Relative separation
d = 2R sin(ωt)ˆi ⇒ ~v A/B
d = 2Rω cos(ωt)ˆi
Relative velocity ~v A
− ~v B
= ~v A/B
Polar coordinate : position, velocity, acceleration
ˆr , ϕˆ different from point to point – change with ϕ but not with r
dˆr
dϕ
= ˆϕ,
d ϕˆ
dϕ
= −ˆr ⇒
dˆr
dt
= ˙ϕ ϕ,ˆ
d ϕˆ
dt
= − ϕ˙ ˆr
Position vector
~r = x
i + y
j = r ˆr
Velocity vector
~v =
~r = x˙
i + ˙y
j = r˙ ˆr + r
ˆr = r˙ ˆr + r ϕ˙ ϕˆ
Here, v r
= ˙r is radial velocity and v ϕ
= r ϕ˙ is angular velocity.
Acceleration vector
~a =
~v =
~r = ¨x
i + ¨y
j =
¨r − r ϕ˙
2
ˆr + (r ϕ¨ + 2˙r ϕ˙) ˆϕ
Can you identify the centripetal / centrifugal & coriolis terms?
Coordinate systems : Cylindrical
Cylindrical coordinate : (ˆρ, ϕ,ˆ
k)
Useful when problems have cylindrical symmetry! e.g. motion on
cylinder etc. (quite useful in electromagnetism)
x = ρ cos ϕ
y = ρ sin ϕ
z = z
ρ ˆ = cos ϕ
i + sin ϕ
j
ϕ ˆ = − sin ϕ
i + cos ϕ
j
k =
k
where, ρ ≥ 0 , 0 ≤ ϕ ≤ 2 π, |z| ≥ 0.
Position vector
~r = x
i + y
j + z
k = ρρˆ + z
k
Velocity vector
~v = ρ˙ ρˆ + ρ ϕ˙ ϕˆ + ˙z
k
Acceleration vector
~a =
ρ¨ − ρ ϕ˙
2
ρˆ + (ρ ϕ¨ + 2 ˙ρ ϕ˙) ˆϕ + ¨z
k
