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VECTORS
Most of the quantities measured in science are classified as either scalars or vectors.
Scalar : A scalar quantity is one which has only magnitude but no direction.
Examples : Mass, time, speed, work, energy, volume, density etc. are scalars.
Vectors : A vector quantity is one which has both magnitude and direction.
Examples : Displacement, velocity, momentum, force etc., are vectors.
To describe a scalar two factors are necessary.
(a) The specific unit of that quantity.
(b) The number of times that unit is contained in the quantity.
To describe a vector three factors are necessary.
(a) The specific unit of that quantity.
(b) The number of times that unit is contained in the quantity.
(c) The orientation of that quantity.
A vector is represented by an arrow. The length of the arrow is proportional to the
magnitude of the vector and its orientation gives the direction of the vector.
The magnitude of a vector is called modulus of the vector. The modulus of a vector PQ
is represented by | PQ | and it is always positive.
The vectors of the same physical quantity are equal if they are of the same magnitude
and have the same direction.
Let A and B be two vectors to be added. Now choose any point P and draw PQ and
QR = B such that the terminus of PQ and the origin of QR. Now PR = C is said to be
the sum of A and B.
A+B=C
Addition of vectors
(a) Commutative Law : In adding two vectors A and B the order of addition makes
no difference. This is called commutative law.
A+B=B+A
(b) Associative Law : Suppose there are three vectors A , B and C . If we first add
A and B then add C we get the same result as that obtained by adding A to the
sum (B + C).
(A + B) + C = A + (B + C)
This is called Associative Law.
(c ) Distributive Law : The distributive law of algebra is applicable to multiplication of
Vectors by a scalar.
This means (n + m)P = nP + mP and m(P + Q) = mP + mQ.
The subtraction of vector B from vector A is same as addition of –B to vector A.
Therefore, the direction of vector B is reversed and added to vector A.
A – B = A + (-B)
Resolution of a Vector :
A vector can be resolved into two components in any two directions. That means two
vectors can be found whose vector sum equals the given vector. Those are called its
components and this single (given) vector is called their resultant.
Resolution of a vector
It is convenient to resolve any vector into two components which are at right angles to
each other. Then they are called rectangular components. They are represented by the
two adjacent sides of a rectangle in which the diagonal represents the given vector both
in magnitude and direction.
C is the vector whose components along the directions OX and OY are A and B. If C
makes an angle ߠ with OX, then moduli of the component vectors are
|A| = |C| cos ߠ
|B| = |C| sin ߠ
To find the component of C (=OR), parallel lines are drawn to OX and OY through the
two end points of the vector A. For convenience, the beginning of C is taken at O.
Then the parallelogram formed becomes a rectangle OPRQ and components of C are
two adjacent sides OP are PR (=OQ).
The resultant of two vectors like velocities, accelerations, forces etc. can be obtained
analytically by parallelogram law or triangle law.
Parallelogram law :
If two vectors are represented in magnitude and direction by the two adjacent sides of
a parallelogram drawn from a point, their resultant is represented as magnitude and
direction by the diagonal passing through the same point.
Parallelogram law of vectors
Explanation : Let a particle at ‘O’ posseses simultaneously two uniform velocities u
and v represented by the vectors OA and OB respectively. The angle between u and v
is ߠ. Then
R2 = OA2 + AC2 + 2 OA AC cos ߠ
R2 = u2 + v2 + 2uv cos ߠ
R=
The direction of the resultant : Let the resultant R makes an angle (
) with the
direction of velocity u. In the triangle COD,
= tan-1 [
]
Special Cases :
(i)
When the two velocities are in the same direction, ߠ = 0˚; cos ߠ = 1 and
sin ߠ = 0.
Then R =
and
=u+v
= tan-1(0) = 0
Therefore, the resultant is equal to the sum of the two velocities and acts in the
direction of these velocities.
(ii)
When the two velocities are in opposite directions :
ߠ = 180˚;
cos 180˚ = -1
and sin 180˚ = 0
Then R =
and
= u – v or v - u
= 0
Therefore, the resultant is equal to the difference between two velocities and
has direction of the larger velocity.
(iii) When the two velocities are at right angles to one another :
ߠ = 90˚;
cos 90˚ = 0;
R=
and
sin 90˚ = 1
= tan-1
If the two velocities are equal, v=u
R=
and
=
=
×u
= tan-1(1) = 45˚
(iv) If two equal velocities have a resultant equal to either, the angle between them is
obtained as follows :
v = u = R; ߠ = ?
u2 = u2 + u2 + 2u2 cosߠ
cos ߠ = - ; ߠ = 120˚
Triangle Law of Vectors :
If two vectors (velocities, accelerations, forces etc.,) are represented in magnitude and
direction by the sides of a triangle taken in order, the resultant or vector sum is
represented in magnitude and direction by the third side of the triangle taken in the
reverse order.
Polygon Law of Vectors :
If a number of vectors are represented in magnitude and direction by the sides of a
polygon taken in order, the resultant is represented in magnitude and direction by the
closing side of the polygon taken in the reverse order.
Product of Two Vectors :
It should be remembered that vectors being added together must be of the same kind,
i.e., displacement vectors are added to displacement vectors or velocity vectors are
added to velocity vectors.
However, like scalars, vectors of different kinds can be multiplied by one another to
generate quantities of new physical dimensions. Because vectors have direction as well
as magnitude, they do not exactly follow the same rules as the algebraic rules of scalar
multiplication.
The scalar or dot product of two vectors A and B is defined as the product of the
magnitudes of the two vectors and the cosine of the angle between them.
The scalar product is read as A dot B.
A B = |A||B| cos ߠ
or A B = AB cos ߠ
This is equivalent to the product of one vector and the component of the second vector
in the direction of the first.
The dot product of two vectors is a scalar quantity and is given by AB cos ߠ. In the
special case when ߠ = 90˚, it becomes zero.
When ߠ = 0˚, it equals the product of the magnitudes of two vectors. The order of the
factors in a scalar product does not effect the product, since the factor cos ߠ remains
the same in both the cases.
[cos (-ߠ) = cos ߠ]
A B=B A
The Characteristics of a Scalar Product :
(a) Scalar product obeys commutative law
A B=B A
(b) Scalar product obeys distributive law
A (B + C) = A B + A C
(c) The scalar product of two mutually perpendicular vectors is zero.
A B = AB cos ߠ = 0 [When, ߠ = 90˚]
The work done by force ‘F’ is given by
F S = |F||S| cos ߠ = F S cos ߠ
Work done is a scalar. It is the dot product of the two vectors F and S.
The vector product of two vectors A and B of moduli A and B respectively is a vector
whose modulus is AB sin ߠ where ‘ߠ’ is the angle between the two vectors and whose
direction is perpendicular to the plane containing A and B
A×B=C
The modulus of C is given by
|C| = |A||B| sin ߠ
A , B and C form a right-handed coordinated system.
The direction of C can be conveniently obtained by (a)right handed screw rule or (b)right
hand thumb rule.
(a) Right-hand screw rule : Let a right hand screw whose axis is perpendicular to
the plane containing A and B be rotated in the direction from A to B. Then the
direction along with the screw advances gives the direction of the product vector
C.
It may be noted, that if the direction of the rotation of the screw is reversed, the
direction of C is also reversed. Therefore, A × B is not the same as B × A.
Hence A × B
B×A
(b) Right hand thumb rule : Let a right hand be held such that the thumb is erect
and the fingers are folded round. If the direction of rotation of the vector from A to
B is same as the direction of folding of fingers, then the thumb points in the
direction of the product vector C.
Geometric Interpretation of Vector Product of two Vectors :
Consider a parallelogram PQRS such that PQ = A and PS = B. The angle between A
and B is ߠ. ‘h’ is the height of the parallelogram.
|A × B| = (PQ) (PS) sin ߠ
= A B sin ߠ = Ah = base × height ( B sin ߠ = h)
= Area of the parallelogram PQRS
Area of the parallelogram
The modulus of vector product of two vectors is equal to area of the parallelogram with
these vectors as adjacent sides and the direction of resultant vector being perpendicular
to the plane of the parallelogram.
The characteristics of cross product :
(a) Cross product does not obey commutative law.
A×B
B×A
(b) Cross product obeys distributive law of multiplication.
A × (B + C) = A × B + A × C
(c) The cross product of two parallel or anti-parallel vectors is zero.
A × B = A B sin ߠ = 0 ( ߠ = 0)
Applications :
(i) Moment of Force about a point :
If r is the position vector of the point P through which the line of action of the
force F passes and AP = r. Then,
Moment of F about the point A = r × F
Thus a vector rotation can be assigned to the moment of the force called
torque as the cross product of the vectors displacement and the force.
(ii) Angular velocity of a rigid body :
Let a rigid body be rotating about the axis OM with angular velocity
radian/
second. Let r be the position vector of a point P i.e. OP = r and v be the linear
velocity of the point P, then
A = r × a.
Unit Vector :
A vector having unit magnitude is called unit vector.
If |a| = 0, then
, is a unit vector in the direction of a.
Some Characteristics of I, j and k :
If i, j and k are unit vectors along x, y and z axis then
i i=j j=k k=1
i j=j k=k i=0
i×i=j
j=k
i×j=k=-j
k=0
i
j×k=i=-k
j
k×i=j =-i
k
Position Vector :
If (x, y, z) are the coordinates of a point P and O is the origin of coordinates, then
OP = xi + yj + zk
Magnitude of OP =
Unit vector parallel to
OP =
Null Vector or Zero Vector :
A vector whose origin and terminus are the same is called null vector or zero vector.
Left-handed
Right-handed
Its magnitude is zero and direction is indeterminate.
Vectors like torque, angular momentum, angular velocity are called axial vectors or
pseudo vectors. The direction of a pseudo vector is given by right-hand thumb rule. This
rule is quite arbitrary and it is a convention. The direction of rotation, clock-wise anticlockwise, decides the direction of a pseudo vector.
Vectors like displacement, velocity, acceleration, force etc., are called real vectors or
polar vectors. The direction of a polar vector is inherent and it is not decided by any
convention. The direction of a polar vector is independent of the coordinate system.
That is by the transformation of the axis, the direction of the polar vector does not
change. In the case of pseudo vectors whenever the coordinate system is transformed
from right handed reference frame to left handed reference frame, its direction is
reversed. The cross-product of two polar vectors is a pseudo-vectors.