2 Vector
1 Vectors Properties of Vectors
In a vector +ve sign and sign –ve indicate
A vector is a quantity that has both direction only.
magnitude and direction. Vectors are used Ex: +5N and –5N, same magnitude of
to represent physical quantities that cannot force in opposite direction.
be fully described by a single number alone. Angle between vector – When two vectors
Examples of vector quantities include are placed head to head or tail to tail then
smaller angle between vector is called
displacement, velocity, acceleration, and
angle between vector.
force.
Scalar Quantity Vector Quantity
Having Having Magnitude Xβ
θ
Magnitude only and direction
Vector can be shifted parallel to itself by
Follow simple follow triangle law of
keeping magnitude and direction fixed.
algebric vector addition.
Rotation of vector not allowed it will
addition change meaning of vector.
Can be changed Can be changed by
If angle between A and B vector is θ then
only by changing magnitude
changing its only, or changing dirn angle between A and -B is (180°-θ).
value only or changing both. All equal vectors are parallel but all
parallels are not equal.
Ex-Speed, time, Ex-Force, Velocity,
current density, torque All opposite (Negative) Vectors are
Mass, Volume,
Antiparallel but all antiparallel are not
density current, etc. etc.
Opposite Vector
2 Type of Vectors
Type Magnitude Direction\Angle
Equal Vector Same Same (θ = 0)
Parallel Vector May or May not same Same (θ = 0)
Opposite Vector or Negative Vectors Same Opposite θ = 180°
Anti-parallel Vector May or May not same θ = 180° opposite
Orthogonal May same θ = 90°
Zero/Null Vector Zero any direction
A
Unit Vectors One A=
A
, Components of Vector in 2-D (effect of If vector is making an angle α, β and γ
Vector) from x, y and z-axis respectively then
j cos2α + cos2β + cos2γ = 1 ; sin2 α + sin2 β
+ sin2 γ = 2
A Ay = A sin θ Direction Cosine
θ Ax Ay Az
i cos α = cos β = cos γ =
Ax = A cos θ A A A
A = Axi + Ay j
3 Vectors addition
A = Acos θ i + A sin θ j
It is the process of combining two vectors
by placing the tail of one vector at the
B By =B cos θ head of the other. i.e.,
θ
R
Bx = B sin θ
B = = R
A + A
B =Bxi + By j
B
= B sin θ i + B cos θ j
Magnitude of resultant Vector Polygon Law of vector addition
2 2 2 2
A = Ax + Ay or B = Bx + By
Start tail of next vector from head of
Direction: previous vector and so on.
Ay Bx
θ = tan–1 –1
or θ = tan
Ax By C
Rectangular component of a vector in 3D A + B + C= R
R
A = Axi + Ay j + Azk
Magnitude B
2 2 2
A = Ax + Ay + Az
A
z
B A + B + C= O
AZ
C
A A
k Triangle Law of Vector addition
j Ay
i y R
B
Ax B resultant must be
θ α in the plane of
θ
A A and B .
x
18
Physics
1 Vectors Properties of Vectors
In a vector +ve sign and sign –ve indicate
A vector is a quantity that has both direction only.
magnitude and direction. Vectors are used Ex: +5N and –5N, same magnitude of
to represent physical quantities that cannot force in opposite direction.
be fully described by a single number alone. Angle between vector – When two vectors
Examples of vector quantities include are placed head to head or tail to tail then
smaller angle between vector is called
displacement, velocity, acceleration, and
angle between vector.
force.
Scalar Quantity Vector Quantity
Having Having Magnitude Xβ
θ
Magnitude only and direction
Vector can be shifted parallel to itself by
Follow simple follow triangle law of
keeping magnitude and direction fixed.
algebric vector addition.
Rotation of vector not allowed it will
addition change meaning of vector.
Can be changed Can be changed by
If angle between A and B vector is θ then
only by changing magnitude
changing its only, or changing dirn angle between A and -B is (180°-θ).
value only or changing both. All equal vectors are parallel but all
parallels are not equal.
Ex-Speed, time, Ex-Force, Velocity,
current density, torque All opposite (Negative) Vectors are
Mass, Volume,
Antiparallel but all antiparallel are not
density current, etc. etc.
Opposite Vector
2 Type of Vectors
Type Magnitude Direction\Angle
Equal Vector Same Same (θ = 0)
Parallel Vector May or May not same Same (θ = 0)
Opposite Vector or Negative Vectors Same Opposite θ = 180°
Anti-parallel Vector May or May not same θ = 180° opposite
Orthogonal May same θ = 90°
Zero/Null Vector Zero any direction
A
Unit Vectors One A=
A
, Components of Vector in 2-D (effect of If vector is making an angle α, β and γ
Vector) from x, y and z-axis respectively then
j cos2α + cos2β + cos2γ = 1 ; sin2 α + sin2 β
+ sin2 γ = 2
A Ay = A sin θ Direction Cosine
θ Ax Ay Az
i cos α = cos β = cos γ =
Ax = A cos θ A A A
A = Axi + Ay j
3 Vectors addition
A = Acos θ i + A sin θ j
It is the process of combining two vectors
by placing the tail of one vector at the
B By =B cos θ head of the other. i.e.,
θ
R
Bx = B sin θ
B = = R
A + A
B =Bxi + By j
B
= B sin θ i + B cos θ j
Magnitude of resultant Vector Polygon Law of vector addition
2 2 2 2
A = Ax + Ay or B = Bx + By
Start tail of next vector from head of
Direction: previous vector and so on.
Ay Bx
θ = tan–1 –1
or θ = tan
Ax By C
Rectangular component of a vector in 3D A + B + C= R
R
A = Axi + Ay j + Azk
Magnitude B
2 2 2
A = Ax + Ay + Az
A
z
B A + B + C= O
AZ
C
A A
k Triangle Law of Vector addition
j Ay
i y R
B
Ax B resultant must be
θ α in the plane of
θ
A A and B .
x
18
Physics
