, Vector Analysis
[From Basic to Advance]
1. Elementary Concept of Vector:
The physical parameter having both direction and magnitude obeying definite rules of addition
and subtraction is called vector. Basically it is a tensor of rank -1 because it will change for the
change of reference frame and in that case the new vector
can be related with the old vector in previous frame by a
single coefficient. That is why it is a vector of rank-1 because
here the no of coefficient appeared in the relation between
the old and new tensor will be the rank of the corresponding
tensor and here it is 1 for vector. For this reason a vector is
denoted by, say, 𝐀 ⃗ where the magnitude of it is |𝐀 ⃗ | or 𝐀. In
diagram, a vector is represented by an arrow where the
direction of this arrow gives the direction of the given vector
and the length of this arrow gives the magnitude of this given
vector. Since the length of an arrow can never be negative so the magnitude of a vector will be
⃗ | =positive.
always positive, i.e. |𝐀
The basic properties of this vector are
a) Any given vector can mathematically represented by the
product of its magnitude with directional unit vector along it.
⃗
𝐀
⃗ = |𝐀
So for a given vector 𝐀 ⃗ |𝐧
̂ = 𝐀𝐧
̂ where ̂𝐧 =
𝐀
b) By rotating a given vector through 180 angle the new vector obtained is called opposite vector
⃗ is given vector then its opposite vector will be −𝐀
of that given vector. . If 𝐀 ⃗ where we have
⃗ = |𝐀
mathematically 𝐀 ⃗ |𝐧
̂ = 𝐀𝐧
̂ and ⃗ = |𝐀
−𝐀 ⃗ |(−𝐧
̂) = 𝐀 (−𝐧
̂)
c) If a given vector be added with its opposite vector then the resultant vector obtained is called
⃗ + (−𝐀
zero vector or inactive vector. Mathematically this vector is given by 𝐀 ⃗ ) = ⃗𝟎. The basic
characteristics of this zero vector are i) It has magnitude zero but it is significant in vector algebra.
Ii) It is usually represented by a dot and thus the direction of this zero vector is uncertain.
⃗ +𝐁
d) The magnitude of resultant of two vectors is given by 𝐑 = |𝐀 ⃗ | = (𝐀𝟐 + 𝐁𝟐 + 𝟐𝐀𝐁𝐂𝐨𝐬𝛂)𝟏/𝟐
and the magnitude of subtraction of two vectors is given by
⃗ −𝐁
𝐫 = |𝐀 ⃗ | = (𝐀𝟐 + 𝐁 𝟐 − 𝟐𝐀𝐁𝐂𝐨𝐬𝛂)𝟏/𝟐 where two vectors 𝐀 ⃗ and 𝐁 ⃗ have angle 𝛂 between
them and all other notations and symbols have their usual meanings. Here also we should note
that
𝐁.𝐒𝐢𝐧𝛂
⃗⃗ makes angle 𝛉 with vector 𝐀
i) If this resultant vector 𝐑 ⃗ then 𝐭𝐚𝐧𝛉 =
𝐀+𝐁𝐂𝐨𝐬𝛂
𝐁.𝐒𝐢𝐧𝛂
⃗ then 𝐭𝐚𝐧𝛉 =
ii) If this subtraction vector 𝐫 makes angle 𝛉 with vector 𝐀 𝐀−𝐁𝐂𝐨𝐬𝛂
⃗ +𝐁
iii) (𝐀 − 𝐁) ≤ |𝐀 ⃗ | ≤ (𝐀 + 𝐁) and (𝐀 + 𝐁) ≥ |𝐀
⃗ −𝐁
⃗ | ≥ (𝐀 − 𝐁)
, e) The mathematical form of the component of that vector
⃗ 𝐨 = (𝐀𝐂𝐨𝐬𝛉)𝐧
along any given direction 𝐀 ̂ where 𝐧
̂ is the unit
vector along that given direction.
f) Any vector can be presented as the resultant of those two
components in two dimension as
⃗ = (𝐀𝐂𝐨𝐬𝛉)𝐢̂ + (𝐀𝐒𝐢𝐧𝛉)𝐣̂ = 𝐀 𝐱 𝐢̂ + 𝐀 𝐲 𝐣̂
𝐀
Similarly in 3 dimensions, it is expressed as
⃗ = (𝐀𝐂𝐨𝐬𝛂)𝐢̂ + (𝐀𝐂𝐨𝐬𝛃)𝐣̂ + (𝐀𝐂𝐨𝐬𝛄)𝐤
𝐀 ̂
̂
= 𝐀 𝐱 𝐢̂ + 𝐀 𝐲 𝐣̂ + 𝐀𝐳 𝐤,
g) In three dimensions we usually have the position
̂ , where
vector at a given point P is 𝐫 = 𝐱 𝐢̂ + 𝐲𝐣̂ + 𝐳𝐤
|𝐫| = 𝐫 = √𝐱 𝟐 + 𝐲 𝟐 + 𝐳 𝟐 .
h) For addition of more than two vectors, the magnitude of their resultant will be
𝐑 = √𝐑𝟐𝐗 + 𝐑𝟐𝐘 = √(∑ 𝐀𝐢 𝐂𝐨𝐬𝛉𝐢 )𝟐 + (∑ 𝐀𝐢 𝐒𝐢𝐧𝛉𝐢 )𝟐 and if this resultant vector makes angle 𝛗
𝐑𝐘 ∑ 𝐀𝐢 𝐒𝐢𝐧𝛉𝐢 𝐑 ∑ 𝐀 𝐒𝐢𝐧𝛉
with X axis then 𝐭𝐚𝐧𝛗 = 𝐑𝐗
= ∑ 𝐀𝐢 𝐂𝐨𝐬𝛉𝐢
⟹ 𝛗 = 𝐭𝐚𝐧−𝟏 (𝐑 𝐘) = 𝐭𝐚𝐧−𝟏 (∑ 𝐀 𝐢𝐂𝐨𝐬𝛉𝐢 )
𝐗 𝐢 𝐢
2. Invariance of Length of Vector under Rotations:
Here we can show that the magnitude of a given vector remain invariant under rotation. To
establish it, let us consider a position vector in two dimensions which is given by 𝐫 = 𝐱 𝐢̂ + 𝐲𝐣̂ . For
rotation of the coordinate system XY through angle , if that position vector changes from 𝐫 to 𝐫′
⃗⃗ = 𝐱 ′ 𝐢̂ + 𝐲′𝐣̂ we have 𝐱′ 𝐂𝐨𝐬𝛉 𝐒𝐢𝐧𝛉 𝐱
then we for 𝐫′ ( )=( ) ( ) and then
𝐲′ −𝐒𝐢𝐧𝛉 𝐂𝐨𝐬𝛉 𝐲
𝐱 ′ = 𝐱𝐂𝐨𝐬𝛉 + 𝐲𝐒𝐢𝐧𝛉 And 𝐲 ′ = −𝐱𝐒𝐢𝐧𝛉 + 𝐲𝐂𝐨𝐬𝛉
So we have 𝐫′𝟐 = 𝐱′𝟐 + 𝐲′𝟐 = (𝐱𝐂𝐨𝐬𝛉 + 𝐲𝐒𝐢𝐧𝛉 )𝟐 + (−𝐱𝐒𝐢𝐧𝛉 + 𝐲𝐂𝐨𝐬𝛉)𝟐 . So finally we get
𝐫′𝟐 = 𝐱 𝟐 𝐂𝐨𝐬 𝟐 𝛉 + 𝐲 𝟐 𝐒𝐢𝐧𝟐 𝛉 + 𝟐𝐱𝐲𝐂𝐨𝐬𝛉𝐒𝐢𝐧𝛉 + 𝐱 𝟐 𝐒𝐢𝐧𝟐 𝛉 + 𝐲 𝟐 𝐂𝐨𝐬 𝟐 𝛉 − 𝟐𝐱𝐲𝐂𝐨𝐬𝛉𝐒𝐢𝐧𝛉
𝟐
And w e get 𝐫′𝟐 = 𝐱 𝟐 + 𝐲 𝟐 = 𝐫 𝟐 i.e. ⃗⃗ | or, |𝐫| = |𝐫′
|𝐫|𝟐 = |𝐫′ ⃗⃗ | . So the magnitude of a vector
remains invariant under rotation.
3. Scalar and Vector Fields
We know that many physical quantities like temperature, electric or gravitational field,
etc. have different values at different points in space, for example, the electric field of a
point charge is large near the charge and it decreases as we got farther away from the
charge. So we can say that the electric field here is the physical quantity that varies from
point to in space and it can be expressed as a continuous function of the position of a
point in that region of space.
[From Basic to Advance]
1. Elementary Concept of Vector:
The physical parameter having both direction and magnitude obeying definite rules of addition
and subtraction is called vector. Basically it is a tensor of rank -1 because it will change for the
change of reference frame and in that case the new vector
can be related with the old vector in previous frame by a
single coefficient. That is why it is a vector of rank-1 because
here the no of coefficient appeared in the relation between
the old and new tensor will be the rank of the corresponding
tensor and here it is 1 for vector. For this reason a vector is
denoted by, say, 𝐀 ⃗ where the magnitude of it is |𝐀 ⃗ | or 𝐀. In
diagram, a vector is represented by an arrow where the
direction of this arrow gives the direction of the given vector
and the length of this arrow gives the magnitude of this given
vector. Since the length of an arrow can never be negative so the magnitude of a vector will be
⃗ | =positive.
always positive, i.e. |𝐀
The basic properties of this vector are
a) Any given vector can mathematically represented by the
product of its magnitude with directional unit vector along it.
⃗
𝐀
⃗ = |𝐀
So for a given vector 𝐀 ⃗ |𝐧
̂ = 𝐀𝐧
̂ where ̂𝐧 =
𝐀
b) By rotating a given vector through 180 angle the new vector obtained is called opposite vector
⃗ is given vector then its opposite vector will be −𝐀
of that given vector. . If 𝐀 ⃗ where we have
⃗ = |𝐀
mathematically 𝐀 ⃗ |𝐧
̂ = 𝐀𝐧
̂ and ⃗ = |𝐀
−𝐀 ⃗ |(−𝐧
̂) = 𝐀 (−𝐧
̂)
c) If a given vector be added with its opposite vector then the resultant vector obtained is called
⃗ + (−𝐀
zero vector or inactive vector. Mathematically this vector is given by 𝐀 ⃗ ) = ⃗𝟎. The basic
characteristics of this zero vector are i) It has magnitude zero but it is significant in vector algebra.
Ii) It is usually represented by a dot and thus the direction of this zero vector is uncertain.
⃗ +𝐁
d) The magnitude of resultant of two vectors is given by 𝐑 = |𝐀 ⃗ | = (𝐀𝟐 + 𝐁𝟐 + 𝟐𝐀𝐁𝐂𝐨𝐬𝛂)𝟏/𝟐
and the magnitude of subtraction of two vectors is given by
⃗ −𝐁
𝐫 = |𝐀 ⃗ | = (𝐀𝟐 + 𝐁 𝟐 − 𝟐𝐀𝐁𝐂𝐨𝐬𝛂)𝟏/𝟐 where two vectors 𝐀 ⃗ and 𝐁 ⃗ have angle 𝛂 between
them and all other notations and symbols have their usual meanings. Here also we should note
that
𝐁.𝐒𝐢𝐧𝛂
⃗⃗ makes angle 𝛉 with vector 𝐀
i) If this resultant vector 𝐑 ⃗ then 𝐭𝐚𝐧𝛉 =
𝐀+𝐁𝐂𝐨𝐬𝛂
𝐁.𝐒𝐢𝐧𝛂
⃗ then 𝐭𝐚𝐧𝛉 =
ii) If this subtraction vector 𝐫 makes angle 𝛉 with vector 𝐀 𝐀−𝐁𝐂𝐨𝐬𝛂
⃗ +𝐁
iii) (𝐀 − 𝐁) ≤ |𝐀 ⃗ | ≤ (𝐀 + 𝐁) and (𝐀 + 𝐁) ≥ |𝐀
⃗ −𝐁
⃗ | ≥ (𝐀 − 𝐁)
, e) The mathematical form of the component of that vector
⃗ 𝐨 = (𝐀𝐂𝐨𝐬𝛉)𝐧
along any given direction 𝐀 ̂ where 𝐧
̂ is the unit
vector along that given direction.
f) Any vector can be presented as the resultant of those two
components in two dimension as
⃗ = (𝐀𝐂𝐨𝐬𝛉)𝐢̂ + (𝐀𝐒𝐢𝐧𝛉)𝐣̂ = 𝐀 𝐱 𝐢̂ + 𝐀 𝐲 𝐣̂
𝐀
Similarly in 3 dimensions, it is expressed as
⃗ = (𝐀𝐂𝐨𝐬𝛂)𝐢̂ + (𝐀𝐂𝐨𝐬𝛃)𝐣̂ + (𝐀𝐂𝐨𝐬𝛄)𝐤
𝐀 ̂
̂
= 𝐀 𝐱 𝐢̂ + 𝐀 𝐲 𝐣̂ + 𝐀𝐳 𝐤,
g) In three dimensions we usually have the position
̂ , where
vector at a given point P is 𝐫 = 𝐱 𝐢̂ + 𝐲𝐣̂ + 𝐳𝐤
|𝐫| = 𝐫 = √𝐱 𝟐 + 𝐲 𝟐 + 𝐳 𝟐 .
h) For addition of more than two vectors, the magnitude of their resultant will be
𝐑 = √𝐑𝟐𝐗 + 𝐑𝟐𝐘 = √(∑ 𝐀𝐢 𝐂𝐨𝐬𝛉𝐢 )𝟐 + (∑ 𝐀𝐢 𝐒𝐢𝐧𝛉𝐢 )𝟐 and if this resultant vector makes angle 𝛗
𝐑𝐘 ∑ 𝐀𝐢 𝐒𝐢𝐧𝛉𝐢 𝐑 ∑ 𝐀 𝐒𝐢𝐧𝛉
with X axis then 𝐭𝐚𝐧𝛗 = 𝐑𝐗
= ∑ 𝐀𝐢 𝐂𝐨𝐬𝛉𝐢
⟹ 𝛗 = 𝐭𝐚𝐧−𝟏 (𝐑 𝐘) = 𝐭𝐚𝐧−𝟏 (∑ 𝐀 𝐢𝐂𝐨𝐬𝛉𝐢 )
𝐗 𝐢 𝐢
2. Invariance of Length of Vector under Rotations:
Here we can show that the magnitude of a given vector remain invariant under rotation. To
establish it, let us consider a position vector in two dimensions which is given by 𝐫 = 𝐱 𝐢̂ + 𝐲𝐣̂ . For
rotation of the coordinate system XY through angle , if that position vector changes from 𝐫 to 𝐫′
⃗⃗ = 𝐱 ′ 𝐢̂ + 𝐲′𝐣̂ we have 𝐱′ 𝐂𝐨𝐬𝛉 𝐒𝐢𝐧𝛉 𝐱
then we for 𝐫′ ( )=( ) ( ) and then
𝐲′ −𝐒𝐢𝐧𝛉 𝐂𝐨𝐬𝛉 𝐲
𝐱 ′ = 𝐱𝐂𝐨𝐬𝛉 + 𝐲𝐒𝐢𝐧𝛉 And 𝐲 ′ = −𝐱𝐒𝐢𝐧𝛉 + 𝐲𝐂𝐨𝐬𝛉
So we have 𝐫′𝟐 = 𝐱′𝟐 + 𝐲′𝟐 = (𝐱𝐂𝐨𝐬𝛉 + 𝐲𝐒𝐢𝐧𝛉 )𝟐 + (−𝐱𝐒𝐢𝐧𝛉 + 𝐲𝐂𝐨𝐬𝛉)𝟐 . So finally we get
𝐫′𝟐 = 𝐱 𝟐 𝐂𝐨𝐬 𝟐 𝛉 + 𝐲 𝟐 𝐒𝐢𝐧𝟐 𝛉 + 𝟐𝐱𝐲𝐂𝐨𝐬𝛉𝐒𝐢𝐧𝛉 + 𝐱 𝟐 𝐒𝐢𝐧𝟐 𝛉 + 𝐲 𝟐 𝐂𝐨𝐬 𝟐 𝛉 − 𝟐𝐱𝐲𝐂𝐨𝐬𝛉𝐒𝐢𝐧𝛉
𝟐
And w e get 𝐫′𝟐 = 𝐱 𝟐 + 𝐲 𝟐 = 𝐫 𝟐 i.e. ⃗⃗ | or, |𝐫| = |𝐫′
|𝐫|𝟐 = |𝐫′ ⃗⃗ | . So the magnitude of a vector
remains invariant under rotation.
3. Scalar and Vector Fields
We know that many physical quantities like temperature, electric or gravitational field,
etc. have different values at different points in space, for example, the electric field of a
point charge is large near the charge and it decreases as we got farther away from the
charge. So we can say that the electric field here is the physical quantity that varies from
point to in space and it can be expressed as a continuous function of the position of a
point in that region of space.
