50 CBSE New Pattern ~ Physics 11th (Term-I)
Motion in a Plane
Quick Revision
1. Scalar Quantity is the physical quantity which 5. Modulus of a Vector The magnitude of a
has only magnitude but no direction. It is vector is called modulus of vector. For a
specified completely by a single number, vector A, it is represented by | A | or A.
alongwith the proper unit. 6. Unit Vector It is a vector having unit
e.g. Temperature, mass, length, time, work, etc. $
magnitude. A unit vector of A is written as A.
2. Vector Quantity is the physical quantity It is expressed as
which has both magnitude and direction and $ = A =A
obeys the triangle/ parallelogram laws of vector A
| A| A
addition and subtraction.
or A = AA $
e.g. Displacement, acceleration, velocity,
momentum, force, etc. In cartesian coordinates, $i, $j and k$ are the unit
3. Representation of Vector A vector is vectors along X -axis, Y -axis and Z -axis.
represented by a bold face type or by
an arrow placed over a letter, It has no unit or dimensions.
® ® ® 7. Equal Vectors Two vectors are said to be
i.e. F , a, b or F , a , b .
equal, if they have equal magnitude and same
The length of the line gives the magnitude and direction.
the arrowhead gives the direction. 8. Resultant Vector It is the combination of
4. Types of Vectors Vectors are classified into two or more vectors and it produces the same
two types polar and axial vectors. effect as two or more vectors collectively
● Polar Vectors Vectors which have a starting produce.
point or a point of application are called polar Two cases for resultant vectors are as follows
vectors. e.g. Force, displacement, etc. Case I When two vectors are acting in
● Axial Vectors Vectors which represent the same direction
the rotational effect and act along the axis of A
rotation are called axial vectors.
e.g. Angular velocity, angular momentum, B
torque, etc.
Resultant vector, R = A + B
, Case II When two vectors are acting in The resultant vector formed in this method is
mutually opposite directions also same as that formed in triangle law of
A addition. i.e. Resultant vector, R = A + B
● Polygon Law of Addition of Vectors This
law states that, when the number of vectors
B
are represented in both magnitude and
Resultant vector, R = A – B direction by the sides of an open polygon
(i) If B > A, then direction of R is along B. taken in an order, then their resultant is
represented in both magnitude and direction
(ii) If A > B, then direction of R is along A. by the closing side of the polygon taken in
9. Addition of Two Vectors (Graphical Method) opposite order.
Two vectors can be added, if both of them are Consider a number of vectors A, B, C and D
of same nature. Graphical method of addition be acting in different directions as shown
of vectors helps us in visualising the vectors D
S
and the resultant vector. T C
This method contains following laws C
+
● Triangle Law of Vector Addition This B Q
R +
law states that, if two vectors can be A B
A+
represented both in magnitude and direction B
by two sides of a triangle taken in the same
O P
order, then their resultant is represented A
completely, both in magnitude and direction, According to this law,
by the third side of the triangle taken in the
opposite order. OT = OP + PQ + QS + ST
Addition N \ Resultant vector, R = A + B + C + D
10. Properties of Addition of Vectors
B
A+ ● It follows commutative law,
B R= B
i.e. A + B = B + A
A θ ● It follows associative law,
O M
A ( A + B) + C = A + ( B + C )
According to triangle law of vector addition, ● It follows distributive law,
ON = OM + MN l( A + B ) = l A + l B
Resultant vector, R = A + B ● A+0=A
● Parallelogram Law of Addition of Two 11. Subtraction of Two Vectors
Vectors This law states that, if two vectors (Graphical Method) If a vector B is to be
are acting on a particle at the same time be subtracted from vector A, then we have to
represented in magnitude and direction by invert the vector B and then add it with vector
two adjacent sides of a parallelogram drawn A, according to laws of addition of two vectors.
from a point, their resultant vector is
Hence, the subtraction of vector B from a vector
represented in magnitude and direction by
A is expressed as R = A + ( - B ) = A - B
the diagonal of the parallelogram drawn
from the same point. B B
C
B B Subtraction A
B R=
B A+ A
Addition R= –B
O α A
A θ –B
O A
A
, 12. Properties of Subtraction of Vectors Y
● Subtraction of vectors does not follow
commutative law
A-B¹B-A Ay
Ay A
● It does not follow associative law
β
γ α Ax
A - ( B - C ) ¹ ( A - B) - C X
● It follows distributive law Az
Az
l( A - B ) = l A - l B Ax
13. Resolution of Vectors in Plane Z l2 +m2+n2=1
Ax
(In Two-Dimensions) The process of splitting Remember that, cos a = =l
a single vector into two or more vectors in A + A 2y + Az2
2
x
different directions which collectively produce Ay
the same effect as produced by the single cos b = =m
vector alone is known as resolution of a vector. A + A 2y + Az2
2
x
The various vectors into which the single Az
cos g = =n
vector is splitted are known as component Ax2 + A 2y + Az2
vectors.
Any vector r can be expressed as a linear Here, l, m and n are known as direction
combination of two unit vectors $i and $j at right cosines of A.
angle, i.e. r = x i$ + y $j. 15. Addition of Vectors (Analytical Method)
According to triangle law of vector addition,
the resultant ( R ) is given by OQ but in opposite
P (x,y)
j order.
Q
r
θ
O
i R B
\Magnitude of resultant vector = | r | = x 2 + y 2 β θ
O P
N
If q is the inclination of r with X-axis, then A
æy ö Resultant, R = A 2 + B 2 + 2 AB cos q
angle, q = tan -1 ç ÷.
èx ø and direction of resultant R,
14. Resolution of a Space Vector B sin q
tan b =
(In Three-Dimensions) We can resolve a A + B cos q
general vector A into three components along
X, Y and Z-axes in three dimensions (i.e. Regarding the Magnitude of R
space). While resolving we have, ● When q = 0 °, then R = A + B (maximum)
Ax = A cos a , ● When q = 90 ° , then R = A 2 + B 2
A y = A cos b, Az = A cos g ● When q = 180 °, then R = A - B (minimum)
\ Resultant vector, 16. Subtraction of Vectors (Analytical Method)
A = A $i + A $j + A k$ There are two vectors A and B at an angle q. If
x y z
we have to subtract B from A, then first invert
Magnitude of vector A is A = Ax2 + A y2 + Az2 the vector B and then add with A as shown in
figure.
Motion in a Plane
Quick Revision
1. Scalar Quantity is the physical quantity which 5. Modulus of a Vector The magnitude of a
has only magnitude but no direction. It is vector is called modulus of vector. For a
specified completely by a single number, vector A, it is represented by | A | or A.
alongwith the proper unit. 6. Unit Vector It is a vector having unit
e.g. Temperature, mass, length, time, work, etc. $
magnitude. A unit vector of A is written as A.
2. Vector Quantity is the physical quantity It is expressed as
which has both magnitude and direction and $ = A =A
obeys the triangle/ parallelogram laws of vector A
| A| A
addition and subtraction.
or A = AA $
e.g. Displacement, acceleration, velocity,
momentum, force, etc. In cartesian coordinates, $i, $j and k$ are the unit
3. Representation of Vector A vector is vectors along X -axis, Y -axis and Z -axis.
represented by a bold face type or by
an arrow placed over a letter, It has no unit or dimensions.
® ® ® 7. Equal Vectors Two vectors are said to be
i.e. F , a, b or F , a , b .
equal, if they have equal magnitude and same
The length of the line gives the magnitude and direction.
the arrowhead gives the direction. 8. Resultant Vector It is the combination of
4. Types of Vectors Vectors are classified into two or more vectors and it produces the same
two types polar and axial vectors. effect as two or more vectors collectively
● Polar Vectors Vectors which have a starting produce.
point or a point of application are called polar Two cases for resultant vectors are as follows
vectors. e.g. Force, displacement, etc. Case I When two vectors are acting in
● Axial Vectors Vectors which represent the same direction
the rotational effect and act along the axis of A
rotation are called axial vectors.
e.g. Angular velocity, angular momentum, B
torque, etc.
Resultant vector, R = A + B
, Case II When two vectors are acting in The resultant vector formed in this method is
mutually opposite directions also same as that formed in triangle law of
A addition. i.e. Resultant vector, R = A + B
● Polygon Law of Addition of Vectors This
law states that, when the number of vectors
B
are represented in both magnitude and
Resultant vector, R = A – B direction by the sides of an open polygon
(i) If B > A, then direction of R is along B. taken in an order, then their resultant is
represented in both magnitude and direction
(ii) If A > B, then direction of R is along A. by the closing side of the polygon taken in
9. Addition of Two Vectors (Graphical Method) opposite order.
Two vectors can be added, if both of them are Consider a number of vectors A, B, C and D
of same nature. Graphical method of addition be acting in different directions as shown
of vectors helps us in visualising the vectors D
S
and the resultant vector. T C
This method contains following laws C
+
● Triangle Law of Vector Addition This B Q
R +
law states that, if two vectors can be A B
A+
represented both in magnitude and direction B
by two sides of a triangle taken in the same
O P
order, then their resultant is represented A
completely, both in magnitude and direction, According to this law,
by the third side of the triangle taken in the
opposite order. OT = OP + PQ + QS + ST
Addition N \ Resultant vector, R = A + B + C + D
10. Properties of Addition of Vectors
B
A+ ● It follows commutative law,
B R= B
i.e. A + B = B + A
A θ ● It follows associative law,
O M
A ( A + B) + C = A + ( B + C )
According to triangle law of vector addition, ● It follows distributive law,
ON = OM + MN l( A + B ) = l A + l B
Resultant vector, R = A + B ● A+0=A
● Parallelogram Law of Addition of Two 11. Subtraction of Two Vectors
Vectors This law states that, if two vectors (Graphical Method) If a vector B is to be
are acting on a particle at the same time be subtracted from vector A, then we have to
represented in magnitude and direction by invert the vector B and then add it with vector
two adjacent sides of a parallelogram drawn A, according to laws of addition of two vectors.
from a point, their resultant vector is
Hence, the subtraction of vector B from a vector
represented in magnitude and direction by
A is expressed as R = A + ( - B ) = A - B
the diagonal of the parallelogram drawn
from the same point. B B
C
B B Subtraction A
B R=
B A+ A
Addition R= –B
O α A
A θ –B
O A
A
, 12. Properties of Subtraction of Vectors Y
● Subtraction of vectors does not follow
commutative law
A-B¹B-A Ay
Ay A
● It does not follow associative law
β
γ α Ax
A - ( B - C ) ¹ ( A - B) - C X
● It follows distributive law Az
Az
l( A - B ) = l A - l B Ax
13. Resolution of Vectors in Plane Z l2 +m2+n2=1
Ax
(In Two-Dimensions) The process of splitting Remember that, cos a = =l
a single vector into two or more vectors in A + A 2y + Az2
2
x
different directions which collectively produce Ay
the same effect as produced by the single cos b = =m
vector alone is known as resolution of a vector. A + A 2y + Az2
2
x
The various vectors into which the single Az
cos g = =n
vector is splitted are known as component Ax2 + A 2y + Az2
vectors.
Any vector r can be expressed as a linear Here, l, m and n are known as direction
combination of two unit vectors $i and $j at right cosines of A.
angle, i.e. r = x i$ + y $j. 15. Addition of Vectors (Analytical Method)
According to triangle law of vector addition,
the resultant ( R ) is given by OQ but in opposite
P (x,y)
j order.
Q
r
θ
O
i R B
\Magnitude of resultant vector = | r | = x 2 + y 2 β θ
O P
N
If q is the inclination of r with X-axis, then A
æy ö Resultant, R = A 2 + B 2 + 2 AB cos q
angle, q = tan -1 ç ÷.
èx ø and direction of resultant R,
14. Resolution of a Space Vector B sin q
tan b =
(In Three-Dimensions) We can resolve a A + B cos q
general vector A into three components along
X, Y and Z-axes in three dimensions (i.e. Regarding the Magnitude of R
space). While resolving we have, ● When q = 0 °, then R = A + B (maximum)
Ax = A cos a , ● When q = 90 ° , then R = A 2 + B 2
A y = A cos b, Az = A cos g ● When q = 180 °, then R = A - B (minimum)
\ Resultant vector, 16. Subtraction of Vectors (Analytical Method)
A = A $i + A $j + A k$ There are two vectors A and B at an angle q. If
x y z
we have to subtract B from A, then first invert
Magnitude of vector A is A = Ax2 + A y2 + Az2 the vector B and then add with A as shown in
figure.
