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Vectors Physics Class 11 - IIT JEE | NEET

eSaral Provides free detailed Vector Physics Notes that will help you in exams like IIT JEE, NEET and Board Preparation. Vector in physics is a quantity that has both magnitude and direction.

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Vectors Physics IIT JEE Notes PDF

Vectors Physics IIT JEE Notes PDF includes all-important formulas, theorems, facts, and equations of vectors. Vectors Physics IIT JEE notes PDF is beneficial for students who are preparing for the JEE exam. Referring to these notes will help them to have a complete revision before the exam. It also helps students to memorise all the important concepts and facts about vectors.

In Physics, we classify quantities into vectors and scalars. The quantities which have both magnitude and direction are called vectors. Examples are velocity, force, displacement, weight, acceleration, etc. The quantities which have only magnitude and no direction are called scalar quantities. Examples are mass, volume, speed, time, frequency, etc.

Click here to download the PDF on Vector Physics IIT JEE Notes

A quantity is said to be a vector if it satisfies the following conditions:

(a) Obeys the law of parallelogram addition.

(b) It has a specified direction.

(c) The addition is commutative, i.e., A+B = B+A

Table of Contents

Representation of a Vector

Position vectors.

• Dot Product of Two vectors

Cross Product of Two Vectors

Projection of a vector, solved examples, practice problems, frequently asked questions.

A vector is represented by a line with an arrowhead. The point O from which the arrow starts is called the tail or initial point, or origin of the vector. Point A, where the arrow ends, is called the tip or head or terminal point of the vector. A vector displaced parallel to itself remains unchanged. If a vector is rotated through an angle other than 360 0 , it changes.

A vector can be replaced by another when its direction and magnitude are the same.

Unit Vector

A vector having a unit magnitude is called a unit vector. It is used to denote the direction of a given vector. \(\begin{array}{l}\vec{A}=\hat{a}A\end{array} \) \(\begin{array}{l}\hat{a}\ \text{is the unit vector along the direction of}\ \vec{A}\end{array} \) .

Types of Vectors

(i) Negative of a vector: It has the same magnitude but opposite direction of the given vector.

(ii) Equal vectors: If two vectors have equal magnitude and direction, they are equal vectors.

(iii) Collinear vectors: Two vectors acting along the same straight lines or along parallel straight lines in the same direction or in the opposite direction are called collinear vectors.

(iv) Coplanar vectors: If three or more vectors lie in the same plane, then they are called coplanar vectors.

(v) Zero vector: It is a vector with zero magnitude and no specific direction.

Addition of Vectors

Law of triangle: If two sides of a triangle are shown by two continuous vectors (vector A and vector B), then the third side of the triangle in the opposite direction shows the resultant of two vectors (vector C).

Vector addition is commutative.

⇒ A + B = B + A

Vector addition is associative.

⇒ A+(B+C) = (A+B)+C

If all sides of a polygon are represented by continuous vectors, the vector sum of all sides is zero.

Polygon method: We use this method when we have to add more than two vectors. It is an extension of the triangular law of vector addition. If a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

Subtraction of vectors

While doing the subtraction of vectors, we change the direction of the vector to be subtracted and then add.

Null vector

If vector A is multiplied by zero, we get a vector whose magnitude is zero, called a null vector or zero vector. The unit of the vector does not change on being multiplied by a dimensionless scalar.

Properties of null vector

1) It has an arbitrary direction.

2) It is represented by a point.

3) It has zero magnitude.

4) Dot product of a null vector with any vector is always zero.

5) Cross product of a null vector with any vector is also a null vector.

6) When a null vector is added or subtracted from a given vector, the resultant vector is the same as the given vector.

Orthogonal unit vectors

The unit vectors along the X-axis, Y-axis, and Z-axis of the right-handed cartesian coordinate system are written as \(\begin{array}{l}\hat{i},\hat{j}\ \text{and}\ \hat{k}\end{array} \) , respectively. They are known as orthogonal unit vectors.

Components of a vector

Consider a vector, V. The components of a vector in a 2D coordinate system are considered to be x-component and y-component. We can represent V = (v x , v y ). Let θ be the angle formed between the vector V and the x-component of the vector. The vector V and its x-component (v x ) form a right-angled triangle if we draw a line parallel to the y-component (v y ).

The horizontal component v x = V cos θ.

The vertical component v y = V sin θ.

A position vector is a vector that gives the position of a point with respect to the origin of the coordinate system. The magnitude of the position vector is the distance of the point P from the origin O. Vector OP is the position vector that gives the position of the particle with reference to O.

Consider point P, whose coordinates are (x, y).

x = r cos θ, y = r sin θ

Magnitude of vector OP is: \(\begin{array}{l}\left | \overrightarrow{OP}\right |=\sqrt{x^{2}+y^{2}+z^{2}}\end{array} \) .

Parallelogram Law of Vectors

If two vectors act along two adjacent sides of a parallelogram having a magnitude equal to the length of the sides, both pointing away from the common vertex, then the resultant is given by the diagonal of the parallelogram passing through the same common vertex and in the same sense as the two vectors.

Dot Product of Two Vectors

If vector A and vector B are two given vectors and θ is the angle between them, then

A and B are the magnitudes of vector A and vector B.

AB cos θ is a scalar quantity. B cos θ is the component of vector B in the direction of vector A.

The dot product of two vectors is the product of the magnitude of one vector with the resolved component of the other in the direction of the first vector.

The dot product is also called a scalar product.

Properties of Dot Product

1) Dot product of two vectors is commutative,

2) The dot product of a vector to itself is the magnitude squared of the vector.

3) The dot product of two mutually perpendicular vectors is zero.

4) Dot product is distributive.

5) The scalar product of two parallel vectors is equal to the product of their magnitudes.

The cross product of two vectors, A and B, is denoted by A × B. Its resultant vector is perpendicular to A and B. Cross products are also called vector products. The cross product of two vectors will give the resultant a vector and can be calculated using the right-hand rule.

Consider two vectors, A and B. The cross product of A and B is a vector having a magnitude equal to the product of the magnitudes of the two vectors and the sine of the angle between them and having the direction perpendicular to the plane containing these vectors. θ is the angle between them, and \(\begin{array}{l}\hat{n}\end{array} \) is the unit vector perpendicular to the plane of vector A and vector B.

Properties of Cross Product

(a) Cross product is not commutative. Consider two vectors, A and B .

(b) Cross product is distributive. Consider three vectors, A , B , and C .

A × ( B + C ) = A × B + A × C

(c) The cross product of two parallel vectors is a zero vector.

For parallel vectors, θ = 0. So, sin θ = 0.

Thus, A × B = 0

(d) The cross product of a vector by itself is a null vector.

(e) The magnitude of the cross product of 2 vectors that are at right angles is equal to the product of the vectors.

(f) The cross product of unit orthogonal vectors, i, j, k, is as follows:

(g) The vector product can be expressed in terms of determinants.

Points to remember

(1) The sum of three non-coplanar forces cannot be zero.

(2) The minimum number of equal forces required for a zero resultant is two.

(3) The minimum number of unequal forces required for a zero resultant is three.

(4) Multiplication of velocity vector by time gives the displacement.

(5) The unit of a vector is not changed if it is multiplied by a dimensionless scalar.

Question 1. Given A = 2i + 3j and B = i + j. The component of vector A along vector B is

The component of vector A along B is A.B/|B|

= (2i + 3j).(i + j)/√(1 + 1)

Hence, option c is the answer.

Question 2. A force is inclined at an angle of 60 °  to the horizontal. If its rectangular component in the horizontal direction is 50 N, then the magnitude of the force in the vertical direction is

Given the horizontal component of force, f x = 50 N

Given angle = 60 0 .

tan 60 = f x /f y

√3 = 50/f y

So f y = 50√3

Question 3. How many minimum number of coplanar vectors having different magnitudes can be added to give zero resultant?

According to the Triangle Law of vector addition, a minimum of three vectors is needed to get zero resultant. A minimum of 3 coplanar vectors is required to represent the same physical quantity with different magnitudes that can be added to give zero results.

Hence, option b is the answer.

Question 4. The square of the resultant of two equal forces is three times their product. The angle between the forces is

Let A and B be the two forces and θ be the angle between them.

Also, A = B.

Given the square of the resultant of two equal forces is three times their product.

F 2 res = 3AB

⇒ F 2 res = A 2 + B 2 + 2AB cos θ

⇒ 3AB = A 2 + B 2 + 2AB cos θ

Since A = B,

⇒ 3A 2 = 2A 2 + 2A 2 cos θ

⇒ A 2 = 2A 2 cos θ

⇒ cos θ = 1/2

Question 5. If A = B + C and the values of A, B, and C are 13, 12, and 5, respectively, then the angle between A and C will be

(a) cos -1 (5/13)

(b) cos -1 (13/12)

(d) sin -1 (5/12)

Given A = B + C

Here, 13 2 = 12 2 + 5 2 .

So, according to the Pythagoras theorem, the angle between B and C is 90 ° .

The angle between A and C = cos θ = 5/13.

⇒ θ = cos -1 (5/13)

Hence, option a is the answer.

Question 6: If the scalar and vector products of two vectors, A and B, are equal in magnitude, then the angle between the two vectors is

Given A.B = A × B

⇒ AB cos θ = AB sin θ

⇒ tan θ = 1

Question 7: If the angle between vectors A and B is θ, then the value of the product (B × A).A is equal to

(a) BA 2 cos θ

(b) BA 2 sin θ

(d) BA 2 sin θ cos θ

B × A will be perpendicular to both A and B.

(B × A).A = (B × A)A cos θ (here θ = 90 ° )

= |B × A||A| cos 90 °

1. If the angle between two vectors A and B is 120 0 , then its resultant C will be

(a) C = |A – B|

(b) C < |A – B|

(c) C > |A – B|

(d) C = |A + B|

2. Three concurrent coplanar forces, 1 N, 2 N, and 3 N, acting along different directions on a body

(a) can keep the body in equilibrium if 2 N and 3 N act at right angles

(b) can keep the body in equilibrium if 1 N and 2 N act at right angles

(c) cannot keep the body in equilibrium

(d) can keep the body in equilibrium if 1 N and 32 N act at right angles

3. Vector A has a magnitude of 5 units and lies in the XY-plane, and points in a direction 120 0 from the direction of increasing X. Vector B has a magnitude of 9 units and points along the Z-axis. The magnitude of cross product A x B is

4. If a.b = |a×b|, then θ will be

5. The direction of A is vertically upward, and the direction of B is in the North direction. The direction of A × B will be

(a) western direction

(b) eastern direction

(c) vertically downward

(d) at 45 0 upward north direction

6. A motorboat covers a given distance in 6 h moving downstream on a river. It covers the same distance in 10 h moving upstream. The time it takes to cover the same distance in still water is

Vectors Class 11 Notes for JEE

State the Polygon law of vector addition.

The Polygon law of vector addition states that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

Define vectors and scalars. Give two examples of each.

Vectors are quantities that have both magnitude and direction. Examples are displacement, weight, acceleration, etc. Scalars are quantities that have only magnitude and no direction. Examples are speed, time, frequency, etc.

What are equal vectors?

Vectors that have the same magnitude and the same direction are called equal vectors. These vectors may start at different positions.

State the parallelogram law of vector addition.

The parallelogram law of vector addition states that if two vectors act along two adjacent sides of a parallelogram having magnitudes equal to the length of the sides, both pointing away from the common vertex, then the resultant is represented by the diagonal of the parallelogram passing through the same common vertex and in the same sense as the two vectors.

What are the different types of vectors?

The different vectors are unit vector, zero vector, equal vector, collinear vector, co-planar vector, position vector, co-initial vector, and like and unlike vectors.

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Coulomb’s Law of Electrostatics

Electrostatic force of interaction acting between two stationary charges is given by

F = 1 / 4π ε o q1q2 / r2

where q1, q2 are magnitude of point charges, r is the distance between them and εo is permittivity of free space.

Here, 1 / 4πε o = (10-7 N – s 2 / C 2 )C 2

Substituting value of c = 2.99792458 X 108 m/s, We get 1 / 4πεo = 8.99 x 109N-m2/C2 In examples and problems we will often use the approximate value,

1 / 4πεo = 9 * 10 9 N-m 2 /C 2 The value of εo is 8.85 * 10 -12 C 2 / N-mC 2 . If there is another medium between the point charges except air or vacuum, then εo is replaced by εoK or εoεr or ε.

where K or εr is called dielectric constant or relative permittivity of the medium. K = εr = ε / εo where, ε = permittivity of the medium.

For air or vacuum, K = 1 For water, K = 81 For metals, K = ∞

Coulomb’s Law in Vector Form

Force on q2 due to q1,

The above equations give the Coulomb’s law in vector form.

Force on q1 due to q2 = – Force on q2 due to q1

F12 = – F21 F12 = q1q2 / 4πε . r1 – r2 / |r1 – r2|3

The forces due to two point charges are parallel to the line joining point charges; such forces are called central forces and electrostatic forces are conservative forces.

Electric Field

The space in the surrounding of any charge in which its influence can be experienced by other charges is called electric field.

Electric Field Lines

An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric field vector at that point. The relative closeness of the lines at some place give an idea about the intensity of electric field at that point.”

Two lines can never intersect.

Electric field lines always begin on a positive charge and end on a negative charge and do not start or stop in mid space.

Electric Field Intensity (E)

The electrostatic force acting per unit positive charge on a point in electric field is called electric field intensity at that point.

Electric field intensity E = Its SI unit is NC-1 or Vim and its dimension is [MLT-3 A-1]. It is a vector quantity and its direction is in the direction of electrostatic force acting on positive charge. Electric field intensity due to a point charge q at a distance r is given by E = 1 / 4π εo q / r2

Electric Potential (V)

Electric potential at any point is equal to the work done per positive charge in carrying it from infinity to that point in electric field. Electric potential, V = W / q Its SI unit is J / C or volt and its dimension is [ML2T-3A-1]. It is a scalar quantity. Electric potential due to a point charge at a distance r is given by v = 1 / 4π εo q / r

Potential Gradient

The rate of change of potential with distance in electric field is called potential gradient.

Potential gradient = dV / dr

Its unit is V / m.

Relation between potential gradient and electric field intensity is given by

E = – (dV / dr)

Equipotential surface is an imaginary surface joining the points of same potential in an electric field. So, we can say that the potential difference between any two points on an equipotential surface is zero. The electric lines of force at each point of an equipotential surface are normal to the surface.

Electric Lines of Force

Electric flux (φe).

Electric flux over an area is equal to the total number of electric field lines crossing this area. Electric flux through a small area element dS is given by

φE = E. dS

where E= electric field intensity and dS = area vector.

Its SI unit is N – m2C-1.

Gauss’s Theorem

The electric flux over any closed surface is 1 / εo times the total charge enclosed by that surface, i.e.,

CBSE Class 11 Physics Notes Electrostatics

If a charge q is placed at the centre of a cube, then

total electric flux linked with the whole cube = q / εo

electric flux linked with one face of the cube = q / 6 εo

(i) Electric Field at Any Point on the Axis of a Uniformly Charged Ring A ring-shaped conductor with radius a carries total charge Q uniformly distributed around it. Let us calculate the electric field at a point P that lies on the axis of the ring at distance x from its centre.

Ex = 1 / 4π εo * xQ / (x2 + a2)3/2

The maximum value of electric field

Ex = 1 / 4π εo (2Q / 3√3R2)

Electric Potential due to a Charged Conducting Spherical Shell

(a) At an extreme point (r > R)

V = 1 / 4π εo q / r

(b) At the surface of a shell (r = R)

V = 1 / 4π εo q / R

(c) At an internal point (r < R)

Therefore potential inside a charged conducting spherical shell equal to the potential at its surface.

(iv) Electric Field and Potential due to a Charged Non-Conducting Sphere

(v) Electric Field Intensity due to an Infinite Line Charge

E = 1 / 2 π εo λ / r

where λ is linear charge density and r is distance from the line charge.

(vi) Electric Field Near an Infinite Plane Sheet of Charge

E = σ / 2 εo

where σ = surface charge density.

If infinite plane sheet has uniform thickness, then

E = σ / εo

An electric dipole consists of two equal and opposite point charges separated by a very small distance. e.g., a molecule of HCL, a molecule of water etc.

• Electric Dipole Moment p = q * 2 l
• Its SI unit is ‘coulomb-metre’ and its dimension is [LTA).
• It is a vector quantity and its direction is from negative charge towards positive charge.

Electric Field Intensity and Potential due to an Electric Dipole

(i) On Axial Line

• Electric field intensity E = 1 / 4 π ε o * 2 pr / (r2 – a2) 2
• If r > > 2a, then E = 1 / 4 π ε o * 2 p / r 3
• Electric potential V = 1 / 4 π εo * p / (r 2 – a 2 )

• I fr > > 2a, then V = 1 / 4 π ε o * p / r 2

(ii) On Equatorial Line

• Electric field intensity E = 1 / 4 π ε o * p / (r2 + a2) 3 / 2

• If r > > 2a, then E = 1 / 4 π ε o * p / r 3
• Electric potential V = 0

(iii) At any Point along a Line Making θ Angle with Axis

• Electric field intensity E = 1 / 4 π εo * p √1 + 3 cos2 θ / r 3

• Electric potential V = 1 / 4 π εo * p cos θ / (r 2 – a 2 cos2 θ)
• If r > > 2a, then V = 1 / 4 π εo * p cos θ / r 2
• Torque acting on an electric dipole placed in uniform electric field is given by
• τ = Ep sin θ or τ = p * E
• When θ = 90°, then ‘τmax = Ep

When electric dipole is parallel to electric field, it is in stable equilibrium and when it is anti-parallel to electric field, it is in unstable equilibrium.

• Work done is rotating an electric dipole in a uniform electric field from angle θ1 to θ2 is given by
• W = Ep (cos θ1 – cos θ2)
• If initially it is in the direction of electric field, then work done in rotating through an angle θ, W = Ep (1 – cos θ).

Potential Energy

Potential energy of an electric dipole in a uniform electric field is given by U = – pE cos θ.

Dipole in Non-uniform Electric Field

• When an electric dipole is placed in a non-uniform electric field, then a resultant force as well as a torque act on it.
• Net force on electric dipole = (qE1 – qE2), along the direction of greater e ecmc field intensity.
• Therefore electric dipole undergo rotational as well as linear motion.

Potential Energy of Charge System

Two point charge system, contains charges q1 and q2 separated by a distance r is given by U = 1 / 4 π εo * q1q2 / r

Three point charge system

U = 1 / 4 π ε o * [q 12 / r 1 + q 2 q 3 / r 2 + q 3 q 2 / r 3

Important Points

• When charge is given to a soap bubble its size gets increased.
• In equilibrium for a charged soap bubble, pressure due to surface tension
• = electric pressure due to charging
• 4T / r = σ 2 / 2 ε o
• or 4T / r = 1 / 2 ε o (q / 4 πr 2 ) 2
• or q = 8 πr √2 ε o r T
• where, r is radius of soap bubble and T is surface tension of soap bubble.

Behaviour of a Conductor in an Electrostatic Field

• Electric field at any point inside the conductor is zero.
• Electric field at any point on the surface of charged conductor is directly proportional to the surface density of charge at that point, but electric potential does not depend upon the surface density of charge.
• Electric potential at any point inside the conductor is constant and equal to potential.

Electrostatic Shielding

• The process of protecting certain field from external electric field is called, electrostatic shielding.
• Electrostatic shielding is achieved by enclosing that region in a closed metallic chamber.

Dielectrics are of two types Non-polar Dielectric The non-polar dielectrics (like N2, O2, benzene, methane) etc. are made up of non-polar atoms/molecules, in which the centre of positive charge coincides with the centre of negative charge of the atom/molecule.

Polar Dielectric

The polar dielectric (like H2O, CO2, NH3 etc) are made up of polar atoms/molecules, in which the centre of positive charge does not coincide with the centre of negative charge of the atom.

• A capacitor is a device which is used to store huge charge over it, without changing its dimensions.
• When an earthed conductor is placed near a charged conductor, then it decreases its potential and therefore more charge can be stored over it.
• A capacitor is a pair of two conductors of any shape, close to each other and have equal and opposite charges.
• Capacitance of a conductor C = q / V
• Its 81 unit is coulomb/volt or farad.
• Its other units are 1 μ F = 10 -6 F
• 1 μμ F = 1 pF = 10 -12 F
• Its dimensional formula is [M -1 L -2 T 4 A 2 ].

Capacitance of an Isolated Spherical Conductor

• C = 4 π ε o K R

• For air K = 1
• ∴ C = 4 π ε o R = R / 9 * 109

Parallel Plate Capacitor

• The capacitor with two plates arranges in parallel is shown below.

• The first plate in the capacitor carries ‘+Q’ charge and the second plate carries ‘–Q’ charge. The area between these plates can be denoted with ‘A’ and the distance(d). Here, ‘d’ is smaller than the area of the plates (d < < A). When the whole charge on the first plate is ‘Q’ & ‘A’ is the area of the plate, then the density of surface charge can be derived as
• σ =Q/A
• Similarly, when the whole charge on the second plate is ‘-Q’ & is the area of the plate is ‘A’, then the density of surface charge can be derived as
• σ = -Q/A
• The regions of this capacitor can be divided into three divisions like area1, area2, and area3. Area 1 is left to the plate1, area 2 is between the planes & area 3 is the right of the second plate. The electric field can be calculated in the region around the capacitor. Here, the electric field is consistent & its path is from the +Ve plate to the –Ve plate.
• V = Exd = 1/ε(Qd/A)
• The capacitance of the parallel plate can be derived as C = Q/V = εoA/d
• The capacitance of a parallel plate capacitor with 2 dielectrics is shown below. Each plate area is Am2 and separated with d-meter distance. The two dielectrics are K1 & k2, then the capacitance will be like the following.
• The capacitance of primary half of the capacitor width is d/2 = C1=> K1Aϵ0/ d/2=> 2K1Aϵ0/d
• Similarly, the capacitance of the next half of the capacitor is C2 = 2K2Aϵ0/d
• Once these two capacitors are connected in series then the net capacitance will be
• Ceff= C1C2/C1+C2= 2Aϵ0/d( K1K2/ /K1+K2)

Capacitors in series and parallel combinations

For practical applications , two or more capacitors are often used in combination and their total capacitance C must be known.To find total capacitance of the arrangement of capacitor we would use equation

(i) Parallel combination of capacitors

• Parallel combination of capacitorsA. Similarly left hand side plates of capacitors would also be at same common potential VB.
• Thus in this case potential difference VAB=VA-VB would be same for both the capacitors, and charges
• Q1 and Q2 on both the capacitors are not necessarily equal. So,
• Q1=C1V and Q2=C2V
• Thus charge stored is divided amongst both the capacitors in direct proportion to their capacitance. Total charge on both the capacitors is,

=V(C1+C2) and

• So system is equivalent to a single capacitor of capacitance

When capacitors are connected in parallel their resultant capacitance C is the sum of their individual capacitances.

The value of equivalent capacitance of system is greater then the greatest individual one. If there are number of capacitors connected in parallel then their equivalent capacitance would be

C=C1+C2+ C3...........

Series combination of capacitors

• Both the points A and B are maintained at constant potential difference VAB.
• In series combination of capacitors right hand plate of first capacitor is connected to left hand plate of next capacitor and combination may be extended foe any number of capacitors.
• In series combination of capacitors all the capacitors would have same charge.
• Now potential difference across individual capacitors are given by

V AR =Q/C 1

V RB =Q/C 2

Sum of VAR and VRB would be equal to applied potential difference V so,

V=V AB =V AR +V RB

=Q(1/C 1 + 1/C 2 )

i.e., resultant capacitance of series combination C=Q/V, is the ratio of charge to total potential difference across the two capacitors connected in series.

So, from equation 12 we say that to find resultant capacitance of capacitors connected in series, we need to add reciprocals of their individual capacitances and C is always less then the smallest individual capacitance.

Result in equation 12 can be summarized for any number of capacitors i.e.,

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