L-1 Vectors.pdf

Representation or notation of a vector:
A vector quantity is represented by two ways. e.g.
(i) By putting a line underneath the letter A as Aor
(ii) By putting a arrow over the letter A as 𝐴.
In print, however, it is represented by the boldfaced type letter like A.
The magnitude modulus of a vector which is a scalar quantity is represented
either by|𝐴| or |A|
or simply by letter printed in italic and without arrow like A.
Here , we will follow notation (ii).
PCIU-MAH Website: abadat26.wixsite.com/pciu

Different Types of Vector
Unit Vector:
A vector of unit magnitude is called a unit vector. When a vector is
divided by its magnitude then we get its unit vector either along or parallel to
the vector. It is denoted by 𝐴 ( read as A hat or A caret) for the vector 𝐴 and the
direction of the unit vector is along the direction of 𝐴 . Sometimes it is
represented by small letter, e.g.𝑎.
Let 𝐴 be a vector whose magnitude, A ≠ 0
∴ 𝐴 =
𝐴
𝐴
, is the unit vector along 𝐴
Normally, in expressing a unit vector a ‘hat’ or a ‘caret’ sign is placed on the top
of the small letter, e.g. 𝑖, 𝑎, 𝑛 etc.
Explanation:
Let 𝐴 = 4𝑎, here 4 is the module or magnitude of the vector and 𝑎 is the unit
vector along the direction of 𝐴.

Equal vectors:
Two more vectors having same magnitude and acting in the
same direction are called equal vectors. In fig. P,Q and S are three
equal vectors.
Here 𝑃 = 𝑄 = 𝑆 𝑎𝑛𝑑 𝑃 = 𝑄 = 𝑆 𝑜𝑟 𝑃 = 𝑄 = 𝑆
Negative or Opposite vector:
Two similar vector having same magnitude but acting opposite to each
other is called negative or opposite vector.
In fig .𝐴𝐵 = 𝑃
Negative vector 𝐵𝐴 = −𝑃 , Here, AB = BA

Position vector:
When the position of a vector is uniquely specified with reference to the
origin of a reference frame, that vector is called position vector. Sometimes,
position vector is called radius vector.
In Fig P is a point with respect to the origin 'O' of the Cartesian co- ordinate
system, therefore 𝑂𝑃 is a position vector of the point P. It is denoted by 𝑟 .
So, 𝑂𝑃= 𝑟 .
Like Vector:
Two vectors of same type, parallel to each other and directed along the
same direction are called like vectors.
Example:𝐴 = 2𝐵

Unlike Vector:
Two vectors of same type, parallel but directed opposite to each
other are called unlike vectors.
In fig., 𝐴 𝑎𝑛𝑑 𝐵 are unlike vectors.
Here, 𝐴 = −3𝐵 .
Null or zero vector :
A vector whose magnitude (or modulus) is zero is called a null or
zero vector. It is denoted by 0 𝑜𝑟 0. Null vector can also defined as a
vector whose initial and terminal point coincide.
Rectangular unit vectors:
In three dimensional co-ordinate system unit vectors 𝑖, 𝑗 𝑎𝑛𝑑 𝑘 along
the respective X, Y and Z axis are called rectangular unit vectors. It
is to be noted that X, Y and Z are perpendicular to each other.

Collinear vectors:
If two or more vectors are directed along the same line or parallel to one
another, then the vectors are called collinear vectors.
Coplanar vectors:
If two or more vectors are parallel to (or lie on) the same plane, then the
vectors are called coplanar vectors. Any plane which is parallel to this plane is
called the plane of vectors.
Localized vectors:
If a vector is restricted to pass through a specified point (i.e. a fixed point) then
it is called a localized vector. An example of a localized vector is a force, its
effect generally depends on the point where it acts.

*** State and explain the General law.
General Law:
Statement:
Of the two vectors, the end or head point of the first vector and the initial point or tail
point of the second vector are placed on the same point, then the direction of the
straight line connecting the initial or tail point of the first vector and the end or head
point of the second vector will give the direction of the resultant vector. The length of
that straight line will give the magnitude of the resultant vector.
Explanation:
Suppose two vectors 𝑃 𝑎𝑛𝑑 𝑄 are acting at the same time at the same point we need
to find out the resultant vector 𝑅.
Let the tail point of the vector 𝑄 represented by line BC be placed at the
head or arrow point of the vector 𝑃 represented by line AB. Now, let us draw the line
AC connecting the tail point of 𝑃 and head point of 𝑄 , then the line AC will give the
resultant vector 𝑅. So, the addition of vectors 𝑃 𝑎𝑛𝑑 𝑄 is
𝑅 = 𝑃 + 𝑄 .

*** State and explain the triangle law.
Statement:
If two similar vectors acting at a point can be represented by two consecutive
sides of a triangle taken in order, then the third side will give the resultant vector in
the reverse order.
Explanation:
Suppose two vectors 𝑃 𝑎𝑛𝑑 𝑄 are acting at the same time at the same point we
need to find out the resultant vector 𝑅.
Let 𝐴𝐵 = 𝑃, From the head of B we draw, 𝐵𝐶 = 𝑄 which is parallel to 𝑄.
Now, joining the initial point of 𝑃 and the terminal or end point of 𝑄 let us
complete the triangle ABC. The side 𝐴𝐶 will represent the resultant vector 𝑅 of 𝑃
and 𝑄 both in the magnitude and direction. That means,
𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶
𝑜𝑟, 𝑃 + 𝑄 = 𝑅
Again,
𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶= −𝐶𝐴
𝑜𝑟, 𝐴𝐵 + 𝐵𝐶 + 𝐶𝐴 = 0
If three vectors acting simultaneously at a point are represented by three
sides of a triangle taken in order, then the resultant will be zero.

*** State the Parallelogram Law. Find the magnitude and direction of the
resultant vector.
The Parallelogram Law:
If two similar vectors acting simultaneously at a point can be represented both
in magnitude and direction by two adjacent sides of a parallelogram, then the
diagonal from the point of intersection of these sides gives the resultant vector
both in magnitude and direction.
Explanation:
Suppose, two vectors𝑂𝐴=𝑃 and𝑂𝐶+𝑄 are acting at the same time at point of a
particle at an angle α [Fig. 1.12]. Taking OA and OC as adjacent sides let us
draw a parallelogram OABC and connect OB.
Now, according to the parallelogram law, the diagonal OB drawn from the tail-
points of 𝑃 and 𝑄 represents the resultant vector𝑅 .i.e.,
𝑂𝐴 +𝑂𝐶 =𝑂𝐵
or, 𝑃 + 𝑄 =𝑅 …..…....…….. (1)

Magnitude of the resultant:
Suppose, the magnitude of the resultant is R and ∠AOC = α is an acute angle
[Figure]. Now from point B let us draw normal BN on the extended line of OA which
intersects the line OA at N.
Now, AB and OC are parallel
∴∠AOC = ∠BAN = α
Again, in ∆OBN, ∠ONB = 90°
∴OB²= ON²+BN²
= (OA + AN)²+BN²
=OA²+2OA.AN+AN²+BN²
But, AB²=AN² +BN² [∵∆BNA is a right angled triangle]
Or, OC² = AN² + BN² [∵ AB = OC ]
Now from trigonometry, AN = 𝐴𝐵 𝑐𝑜𝑠 𝛼 = 𝑂𝐶 𝑐𝑜𝑠 𝛼
So, OB² = OA² + OC² + 2𝑂𝐴 . 𝐴𝐵 𝑐𝑜𝑠 𝛼
or, OB² = OA² + OC² + 2𝑂𝐴 . 𝑂𝐶 𝑐𝑜𝑠 𝛼
or, R² = P² + Q² + 2𝑃𝑄 𝑐𝑜𝑠 𝛼
∴ R = 𝑃² + 𝑄² + 2𝑃𝑄 cos 𝛼 ….…..….….….. (2)

Direction of the resultant:
Suppose the resultant R makes an angle 𝜃 with P .
i.e.,∠𝐴𝑂𝐵 = 𝜃
So, in the right angle triangle ∆𝑂𝐵𝑁 ,
𝑡𝑎𝑛 𝜃 =
𝐵𝑁
𝑂𝑁
=
𝐵𝑁
(𝑂𝐴+𝐴𝑁)
=
𝐴𝐵 sin 𝛼
𝑂𝐴+𝐴𝐵 cos 𝛼
[ ∵𝑠𝑖𝑛 𝛼 =
𝐵𝑁
𝐴𝐵
and 𝑐𝑜𝑠 𝛼 =
𝐴𝑁
𝐴𝐵
]
∴ 𝑡𝑎𝑛 𝜃 =
𝑄 sin 𝛼
𝑃+𝑄 cos 𝛼
…..…....…..….….. (3)
Hence, from equation (2) and (3) we get R and 𝜃 respectively.

Special Cases :
If α = 0° , i.e., vectors 𝑃 and 𝑄 act in the same direction, then
𝑅² = 𝑃² + 𝑄² + 2𝑃𝑄 𝑐𝑜𝑠 0°
or, R = 𝑃² + 𝑄² + 2𝑃𝑄 [ ∵ 𝑐𝑜𝑠 0° = 1 ]
or, R = (𝑃 + 𝑄)²
∴ 𝑅 = 𝑃 + 𝑄
And, 𝑡𝑎𝑛𝜃 =
𝑄 sin 0°
𝑃+𝑄 cos 0°
or, 𝑡𝑎𝑛𝜃 = 𝑡𝑎𝑛 0°
∴𝜃 = 0°
So, the magnitude of the resultant of the two vectors acting in the same direction
is the sum of magnitudes of each vector and the direction of the resultant is in
the same direction along which the vectors act.
If, α = 90°, i.e., vectors 𝑃 and 𝑄 are perpendicular to each other , then
R² = P² + Q² + 2PQ 𝑐𝑜𝑠 180°[ ∵𝑐𝑜𝑠 90° = 0 ]
or, R = 𝑃² + 𝑄²
And 𝑡𝑎𝑛 𝜃 =
𝑄 sin 90°
𝑃+𝑄 cos 90°
=
𝑄
𝑃
∴𝜃 = tan−1
(
𝑄
𝑃
)

If α = 180°, i.e., vectors𝑃 +𝑄 are opposite in direction, then
R² = P² + Q² + 2PQ 𝑐𝑜𝑠 180°
or, R = P² + Q² − 2PQ [ ∵𝑐𝑜𝑠 180° = −1 ]
or, R = (P ∼ Q)²
∴ R = P ∼ Q
And 𝑡𝑎𝑛 𝜃 =
𝑄 sin 180°
𝑃+𝑄 cos 180°
=
0
𝑃 −𝑄
= 0
∴𝜃 = 0
i.e., if vectors are opposite in direction, then the magnitude of the resultant
will be the subtraction of the two vectors and the direction of the resultant will
be along the direction of the larger vector . But if the vectors are equal and
opposite, then the resultant will be zero.

*** Two vectors are acting at a point, show that the maximum and minimum
value of the resultant vector are respectively equal to the addition and
subtraction.
Maximum and minimum value of resultant:
Suppose two vectors 𝑃 and 𝑄 are acting simultaneously at a point making an
angle α . Now according to parallelogram law the magnitude of the resultant,
R = 𝑃² + 𝑄² + 2𝑃𝑄 cos 𝛼
From the above equation it is evident that R depends on the angle between 𝑃
and 𝑄 i.e., on α
R will be maximum when 𝑐𝑜𝑠 𝛼 will be maximum i.e., when 𝑐𝑜𝑠 𝛼 = 1 = 𝑐𝑜𝑠 0°
or, 𝛼 = 0°
𝑅(𝑚𝑎𝑥𝑖𝑚𝑢𝑚)= 𝑃² + 𝑄² + 2𝑃𝑄 cos 𝛼
𝑜𝑟, 𝑅(𝑚𝑎𝑥𝑖𝑚𝑢𝑚)= (𝑃 + 𝑄 )²
𝑜𝑟, 𝑅(𝑚𝑎𝑥𝑖𝑚𝑢𝑚)= ( P + Q ) …..…..…..….. (4)
Thus, when two vectors act along the same straight line then the
magnitude of the resultant will be maximum. The magnitude will be the
sum of vectors. In other words, we can say that magnitude of the resultant
cannot be greater than the summation of the vectors.

The magnitude of 𝑅 will be minimum when 𝑐𝑜𝑠 𝛼 is minimum i.e.,
cos 𝛼 = −1 = cos 180° or,𝛼 = 180°,
∴𝑅 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = 𝑃² + 𝑄² + 2𝑃𝑄 cos 180°
𝑜𝑟, 𝑅 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = 𝑃² + 𝑄² − 2𝑃𝑄
𝑜𝑟, 𝑅 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = ( 𝑃 − 𝑄 )²
𝑜𝑟, 𝑅 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = P ∼ Q …..…..…..…..….. (5)
Thus, when two vectors act along the same line but the opposite
direction, the magnitude of the resultant will be minimum. In other
words, the minimum value of magnitude of the resultant of the two
vectors cannot be less than their subtraction.
[N.B.–sign between P and Q means that the larger one should be written
first e.g., suppose Q > P, then P – Q = Q – P ]

*** What is resolution of a vector? Find the equations of the components
when a vector is resolved into two components.
Resolution of vectors and components:
A vector quality can be resolved into two or more vectors in different directions.
The process of resolving a vector into two or more vectors is called resolution of
a vector or vector resolution. Each resolving vector is called component of the
original vector.
Resolution in any two components :
Let 𝑅 be the vector acting at a point O along OB. The line OB represents
𝑅 both in magnitude and direction. The vector 𝑅 is to be resolved into two
components.
Let the lines OA and OC be drawn at point O making angles
respectively 𝛼 and 𝛽 with OB. Let us complete the parallelogram OABC
[Figure]. Let 𝑂𝐴 = 𝑃 and 𝑂𝐶 = 𝑄
Now from the law of parallelogram, we get
𝑂𝐵 = 𝑂𝐴 + 𝑂𝐶 = 𝑃 +𝑄
∴𝑃 and𝑄 are the two components of 𝑅

Component vectors in scalar from:
To express the component vectors in scalar form, let us apply sine law of triangle
and trigonometry into OABC . We get,
𝑂𝐴
sin 𝛽
=
𝐴𝐵
sin 𝛼
=
𝑂𝐵
sin ∠𝑂𝐴𝐵
or,
𝑃
sin 𝛽
=
𝑄
sin 𝛼
=
𝑅
sin{ 180°− 𝛼+ 𝛽 }
or,
𝑃
sin 𝛽
=
𝑄
sin 𝛼
=
𝑅
sin( 𝛼 + 𝛽 )
…..…..…..….. (1)
∴ P =
𝑅 sin 𝛽
sin( 𝛼 + 𝛽 )
…..…..…..….. (2)
and Q =
𝑅 sin 𝛼
sin ( 𝛼+ 𝛽 )
…..…...... (3)
Equations (2) and (3) are the equations of the components of the vector 𝑅 .

Resolution into perpendicular components :
Let 𝑅 be the vector along OB, so that 𝑂𝐵 =𝑅 . Now taking OB as diagonal let
us draw the rectangular OABC [Figure].
Here the components P and Q are perpendicular to one another, i.e., α + β = 90°.
Putting this value in equations (15) and (16) we get,
P =
𝑅 sin 𝛽
sin 90°
= R sin β
= 𝑅 𝑠𝑖𝑛 ( 90° − 𝛼 )
= 𝑅 𝑐𝑜𝑠 𝛼
And Q =
𝑅 sin 𝛼
sin 90°
= 𝑅 𝑠𝑖𝑛 𝛼
P = 𝑅 𝑐𝑜𝑠 𝛼 and Q = 𝑅 𝑠𝑖𝑛 𝛼 …..…..….….. (9)

*** How to follow the of vector resolution in Lawn Roller.
Lawn Roller:
When an object is either pulled or pushed on a plane, a frictional force acts
between the plane and the object. This retards the motion of the object. The
heavier the object, larger is the frictional force. A lawn roller is made moving
either by pushing or pulling.
In case of pushing :
Let the weight of the roller be 𝑊 and applied force on the handle of the roller
be 𝐹 . Let the force 𝐹 acts at O at angle 𝜃 with the horizontal plane [Fig. (a)].
Now, 𝐹 can be resolved at O into two normal components.
The horizontal component of the force = F𝑐𝑜𝑠 𝜃 , which acts along OB in the
forward direction and the vertical component of the force = 𝑖𝑛 𝜃 , whose
direction is along OC. This increases the weight of the roller.
So, total weight of the roller is ( 𝑊 + 𝜃 ). Which is larger than the actual
weight of the roller. Consequently, the frictional force also increases. So, it is
difficult to push the roller.

In case of pulling :
Let, the weight of the roller = 𝑊 and the applied force on the handle =𝐹 .
The force 𝐹 acts at O at an angle 𝜃 along the horizontal line OB [Fig.(b) ].
This force can be resolved into two normal components. The horizontal
component of F is 𝐹 𝑐𝑜𝑠 𝜃 . Due to its action the roller moves in the forward
directions.
The vertical component of 𝐹 is 𝜃 , which acts upward along OD. So
the total weight of the roller decreases. The weight of the roller is
( 𝑊 – 𝐹 𝑠𝑖𝑛 𝜃 ). Which is less than the actual weight of the roller. The frictional
force also decreases, so it becomes easier to pull a roller.
Conclusion: It is easier to pull a lawn roller than to push.

*** Define and explain the scalar product with figure.
Scalar Product:
The multiplication of the vectors by which a scalar quantity is obtained is
called scalar multiplication. The product of the magnitude of two vectors and
the cosine of the smaller angle between them is called the scalar product.
Explanation:
If 𝜃 is the angle between two vectors B, then according to the definition of
scalar product.
Let 𝐴 and 𝐵 vectors are inclined by an angle, 𝜃, then according to the
definition of scalar product.
𝐴 .𝐵 = AB cos 𝜃 (When, 0≤ 𝜃≤ π)

*** Define and explain the vector product with figure.
Vector Product:
The multiplication of two vector by which a vector quantity is obtained is called
vector multiplication. The product of the magnitude of two vectors and the sine
of the smaller angle between them is called the vector product.
Explanation:
Let 𝐴 and 𝐵 vectors are inclined by an angle, 𝜃, then according to the
definition of vector product.
𝐴 × 𝐵 = AB sin𝜃𝑛

*** Write down the distinction between two kinds of vector multiplication.
*** Write down the distinction between Vector or Cross Product and Scalar
or Dot Product.
Distinction between two kinds of vector multiplication
Vector or Cross Product Scalar or Dot Product
1. If the product of two vectors is a
vector quantity then the product is
called vector or cross product.
1. If the product of two vectors is a
scalar quantity then the product is called
scalar or dot product.
2. If 𝑃 and 𝑄 are vectors, then 𝑃 ×
𝑄 = 𝑃𝑄 𝑠𝑖𝑛𝛼 𝜂 and 𝑃 × 𝑄 =
𝜂 𝑃𝑄 𝑠𝑖𝑛𝛼, here 𝛼 = angle between the
two vectors and 𝜂 is a unit vector
perpendicular to the plane containing
𝑃 𝑎𝑛𝑑 𝑄 .
2. If 𝑃 and 𝑄 are vectors, then 𝑃. 𝑄 =
𝑃𝑄 𝑐𝑜𝑠𝛼. Where, 𝛼 is the angle between
𝑃 𝑎𝑛𝑑 𝑄.
3. Vector product of two vectors doesn’t
follow the commutative law e.g., 𝑃 ×
𝑄 = 𝑄 × 𝑃.
3. Scalar product of two vectors doesn’t
follow the commutative law e.g., 𝑃. 𝑄 =
𝑄. 𝑃.
4. If two vectors are parallel to each
other than their vector product is zero
although the magnitudes of the vectors
are not zero.
4. If the two vectors act perpendicular to
each other, then their scalar product is
zero even though the magnitudes of the
vectors are not zero.

L-1 Vectors.pdf

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