Class 11 Physics Vectors Notes

Class 11 Physics Vectors Notes
UNIT 1
CLASS 11 PHYSICS • MECHANICS

Vectors

Chapter 2

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Scalar and Vector Quantities

Scalar Quantity

The physical quantities which have magnitude but not direction are called scalar quantities or scalars. Scalar quantities can be added or subtracted according to the rule of algebra.

Examples: Mass, time, distance, speed, density, energy, temperature, pressure, charge, gravitational potential, electric potential energy, etc.

Vector Quantity

The physical quantities which have both magnitude and direction are called vector quantities or vectors. Vector quantities can be added or subtracted according to the rule of vector addition.

Examples: Displacement, velocity, acceleration, force, area, weight, electric field, magnetic field, gravitational field, momentum, torque, etc.

A vector is graphically represented by a straight line with an arrow at one end. The direction of the arrow represents the direction of the vector and the length of the line represents the magnitude of the vector.

Graphical representation of a vector from O to A O A magnitude direction →
Graphical representation of a vector: length gives magnitude and the arrow gives direction.

Question

Is it necessary that a quantity having both magnitude and direction be always a vector?

Solution: No. For a quantity to be a vector, it should not be added or subtracted according to the rule of algebra. Electric current has both magnitude and direction, yet it can be added according to the rule of algebra. So, it is a scalar quantity.

Types of Vectors

1. Unit Vector

A vector having unit magnitude is called a unit vector. It is denoted by  and is given by:
 = A⃗ / |A⃗| = A⃗ / A

where A = |A⃗| is the magnitude of vector A⃗. A unit vector is generally used to indicate the direction of a vector.

Note: If A⃗ = x î + y ĵ + z k̂, then
|A⃗| = A = √(x2 + y2 + z2)

Find the unit vector of A⃗ = 3î + 4ĵ + 5k̂

Given: A⃗ = 3î + 4ĵ + 5k̂

|A⃗| = A = √(32 + 42 + 52) = √50 = 5√2
 = A⃗ / |A⃗| = (3î + 4ĵ + 5k̂)/(5√2)
∴ Â = (3/(5√2))î + (2√2/5)ĵ + (1/√2)k̂

2. Null Vector

A vector which has direction but no magnitude, i.e. whose magnitude is zero, is called a null vector. The initial point and terminal point of a null vector coincide.

3. Parallel Vectors

Two or more vectors having the same direction are called parallel vectors. In parallel vectors, the angle between two vectors is 0°.

4. Antiparallel Vectors

Two or more vectors having opposite directions are called antiparallel vectors. In antiparallel vectors, the angle between two vectors is 180°.

5. Equal Vectors

Vectors having the same magnitude and direction are called equal vectors.

6. Negative Vectors

Vectors having the same magnitude but acting in opposite directions are called negative vectors.

Parallel, antiparallel, equal and negative vectors Parallel vectors A⃗B⃗ Antiparallel vectors Equal vectors |A⃗| = |B⃗| and same direction Negative vectors same magnitude, opposite direction Angle: parallel = 0° antiparallel = 180°
Vector relationships shown in the source notes.

7. Coplanar Vectors

Two or more vectors lying on the same plane are called coplanar vectors.

8. Co-initial Vectors

Vectors whose initial points are the same are called co-initial vectors.

9. Co-terminal Vectors

Vectors whose end points are the same are called co-terminal vectors.

Coplanar, co-initial and co-terminal vectors Coplanar vectors A⃗B⃗C⃗ Co-initial vectors O Co-terminal vectors O
Coplanar, co-initial and co-terminal vectors.

Addition of Vectors

The source introduces vector addition through the triangle law, parallelogram law and polygon law.

1. Triangle Law of Vector Addition

If two vectors in magnitude and direction are represented by two sides of a triangle taken in order, then the third side of the triangle taken in the opposite order represents the resultant vector.
Triangle law of vector addition θ A⃗ B⃗ R⃗ O X K M
Triangle law of vector addition and the perpendicular used in the derivation.

Let two vectors A⃗ and B⃗ be represented by two sides of the triangle taken in order. Their resultant R⃗ is represented by the third side taken in the opposite order.

Magnitude of the Resultant

From the right-angled construction used in the source:

R2 = (A + B cosθ)2 + (B sinθ)2
= A2 + 2AB cosθ + B2cos2θ + B2sin2θ
= A2 + 2AB cosθ + B2(cos2θ + sin2θ)
R2 = A2 + B2 + 2AB cosθ
R = √(A2 + B2 + 2AB cosθ)

Direction of the Resultant

Let the resultant R⃗ make angle α with vector A⃗.

tanα = (B sinθ)/(A + B cosθ)
α = tan−1[(B sinθ)/(A + B cosθ)]

If the resultant R⃗ makes angle α with vector B⃗, then the source gives:

α = tan−1[(A sinθ)/(B + A cosθ)]

2. Parallelogram Law of Vector Addition

If two vectors in magnitude and direction are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal passing through their point of intersection, both in magnitude and direction.
Parallelogram law of vector addition θ A⃗ B⃗ R⃗ O A B C D
Parallelogram law: the diagonal through the common initial point gives the resultant.

Let vectors A⃗ and B⃗ be represented by adjacent sides OA and OB of parallelogram OACB. According to the parallelogram law, resultant R⃗ is represented by diagonal OC.

Magnitude

R2 = (A + B cosθ)2 + (B sinθ)2
= A2 + 2AB cosθ + B2(cos2θ + sin2θ)
R2 = A2 + B2 + 2AB cosθ
R = √(A2 + B2 + 2AB cosθ)

Direction

Let resultant R⃗ make angle α with vector A⃗.

tanα = (B sinθ)/(A + B cosθ)
α = tan−1[(B sinθ)/(A + B cosθ)]

If resultant R⃗ makes angle α with vector B⃗:

α = tan−1[(A sinθ)/(B + A cosθ)]

Special Cases of Vector Addition

Case I: Two vectors act parallel, θ = 0°

R = √(A2 + B2 + 2AB cos0°)
= √(A2 + B2 + 2AB)
R = A + B
tanα = (B sin0°)/(A + B cos0°) = 0
α = 0°

Thus, when two vectors act parallel, the resultant has magnitude equal to the sum of their magnitudes and direction along A⃗ and B⃗.

Case II: Two vectors act perpendicularly, θ = 90°

R = √(A2 + B2 + 2AB cos90°)
R = √(A2 + B2)
tanα = B/A
α = tan−1(B/A)

Case III: Two vectors act antiparallel, θ = 180°

R = √(A2 + B2 + 2AB cos180°)
= √(A2 + B2 − 2AB)
R = A − B

If A = B, then R = 0. Thus, when two vectors are equal and acting antiparallel, their resultant is zero.

Subtraction of Vectors

For subtraction, vector B⃗ is reversed and added as −B⃗:

R⃗ = A⃗ + (−B⃗)
R = √[A2 + B2 + 2AB cos(180° − θ)]
R = |A⃗ − B⃗| = √(A2 + B2 − 2AB cosθ)
Subtraction of vectors by adding the negative vector A⃗ B⃗ −B⃗ R⃗ = A⃗ − B⃗ θ
Vector subtraction represented as addition of the negative vector.

Solved Questions on Vector Addition

Q.1(A)

Two vectors A⃗ and B⃗ are such that R⃗ = A⃗ + B⃗ and R2 = A2 + B2. What is the angle between A⃗ and B⃗?

R2 = A2 + B2 + 2AB cosθ
R2 = A2 + B2
2AB cosθ = 0 ⇒ cosθ = 0 = cos90°
θ = 90°

Q.1(B)

Two vectors A⃗ and B⃗ are such that R⃗ = A⃗ + B⃗ and R = A + B. Find the angle between them.

R2 = A2 + B2 + 2AB cosθ
(A + B)2 = A2 + B2 + 2AB
cosθ = 1 = cos0°
θ = 0°

Q.1(C)

Two vectors A⃗ and B⃗ are such that R⃗ = A⃗ + B⃗ and R = A − B. Find the angle between them.

R2 = A2 + B2 + 2AB cosθ
R2 = (A − B)2 = A2 + B2 − 2AB
cosθ = −1 = cos180°
θ = 180°

Q.1(D)

Two vectors A⃗ and B⃗ are such that R⃗ = A⃗ + B⃗ and R⃗′ = A⃗ − B⃗, with equal resultant magnitudes. Find the angle between them.

R2 = A2 + B2 + 2AB cosθ
R′2 = A2 + B2 − 2AB cosθ

Comparing the two expressions:

cosθ = −cosθ ⇒ 2cosθ = 0 ⇒ cosθ = 0
θ = 90°

Q.2

Can the sum of two equal vectors be equal to either of the vectors? Explain.

For A = B = R = a:

R2 = A2 + B2 + 2AB cosθ
a2 = a2 + a2 + 2a2cosθ
1 = 2 + 2cosθ
2cosθ = −1 ⇒ cosθ = −1/2 = cos120°
θ = 120°

Thus, when the angle between two equal vectors is 120°, their sum can be equal in magnitude to either vector.

Q.1(E) – as written in the source

The source gives C⃗ = A⃗ − B⃗ and C = A − B, and asks for the angle between A⃗ and B⃗.

C2 = A2 + B2 − 2AB cosθ
C2 = A2 + B2 − 2AB
cosθ = 1 = cos0°
θ = 0°

3. Polygon Law of Vector Addition

If a number of vectors acting at a point can be represented by the sides of a polygon taken in order, then the resultant is represented by the closing side of the polygon taken in the opposite order, in magnitude and direction.
Polygon law of vector addition A⃗ B⃗ C⃗ D⃗ R⃗ O X Y Z W
Polygon law: the closing side taken in the opposite order represents the resultant.

Let vectors A⃗, B⃗, C⃗ and D⃗ acting at a point O be represented by the sides of polygon OXYZWO. The closing side WO represents the resultant R⃗ in magnitude and direction.

Join O to Y and O to Z, representing intermediate vectors E⃗ and F⃗ respectively.

E⃗ = A⃗ + B⃗   …(i)
F⃗ = E⃗ + C⃗   …(ii)
Using (i) in (ii): F⃗ = A⃗ + B⃗ + C⃗   …(iii)
R⃗ = F⃗ + D⃗   …(iv)
R⃗ = A⃗ + B⃗ + C⃗ + D⃗

Question

Two vectors A⃗ and B⃗ have resultant R⃗. If the direction of B⃗ is reversed, the new resultant is S⃗. Show that R2 + S2 = 2(A2 + B2).

In the first condition:

R2 = A2 + B2 + 2AB cosθ   …(i)

When the direction of B⃗ is reversed:

S2 = A2 + B2 − 2AB cosθ   …(ii)

Adding (i) and (ii):

R2 + S2 = 2A2 + 2B2
R2 + S2 = 2(A2 + B2)   proved.

Resolution of Vectors

The process of splitting a vector into its components is called resolution of vectors.

Consider vector A⃗ resolved into two components: Ay along the y-axis and Ax along the x-axis. Let θ be the angle made by A⃗ with the x-axis.

Resolution of a vector into x and y components θ A⃗ Aₓ Aᵧ x y O P N
Resolution of vector A⃗ into Ax and Ay.
sinθ = Ay/A
Ay = A sinθ   …(i)
cosθ = Ax/A
Ax = A cosθ   …(ii)
Ax2 + Ay2 = A2(cos2θ + sin2θ)
A = √(Ax2 + Ay2)   …(iii)
A sinθ / A cosθ = Ay/Ax
tanθ = Ay/Ax   …(iv)

Product of Two Vectors

(a) Scalar Product or Dot Product of Two Vectors

The product of two vectors is said to be a scalar product if the two vectors multiplied together give a scalar quantity.

The scalar or dot product of vectors A⃗ and B⃗ with angle θ between them is denoted by A⃗ · B⃗ and defined by:

A⃗ · B⃗ = AB cosθ
θ = cos−1[(A⃗ · B⃗)/(AB)]

Special Cases

Parallel, θ = 0°
A⃗ · B⃗ = AB cos0° = AB
Perpendicular, θ = 90°
A⃗ · B⃗ = AB cos90° = 0
Antiparallel, θ = 180°
A⃗ · B⃗ = AB cos180° = −AB

Properties of Dot Product

  1. The dot product is commutative:
    A⃗ · B⃗ = B⃗ · A⃗

    Proof: A⃗ · B⃗ = AB cosθ and B⃗ · A⃗ = BA cosθ, therefore A⃗ · B⃗ = B⃗ · A⃗.

  2. The scalar product is distributive:
    A⃗ · (B⃗ + C⃗) = A⃗ · B⃗ + A⃗ · C⃗
  3. The scalar product of a vector with itself gives the square of its magnitude:
    A⃗ · A⃗ = A2

    Proof: A⃗ · A⃗ = AA cos0° = A2.

(b) Vector Product or Cross Product of Two Vectors

The product of two vectors is said to be a vector product if the two vectors multiplied together give a vector quantity.

The vector or cross product of A⃗ and B⃗ with angle θ between them is denoted by A⃗ × B⃗ and defined as:

A⃗ × B⃗ = AB sinθ n̂

where n̂ is a unit vector perpendicular to both A⃗ and B⃗.

|A⃗ × B⃗| = AB sinθ
θ = sin−1[|A⃗ × B⃗|/(AB)]

Special Cases

Parallel, θ = 0°
A⃗ × B⃗ = AB sin0° n̂ = 0
Perpendicular, θ = 90°
A⃗ × B⃗ = AB sin90° n̂ = AB n̂
Antiparallel, θ = 180°
A⃗ × B⃗ = AB sin180° n̂ = 0

Properties of Vector Product

  1. It is anti-commutative:
    A⃗ × B⃗ = −B⃗ × A⃗
  2. It is distributive:
    A⃗ × (B⃗ + C⃗) = A⃗ × B⃗ + A⃗ × C⃗
  3. The cross product of a vector with itself gives a null vector:
    A⃗ × A⃗ = 0⃗

    Proof: A⃗ × A⃗ = AA sin0° n̂ = 0.

Q.4: If A⃗ · B⃗ = 0, what is the angle between A⃗ and B⃗?

A⃗ · B⃗ = AB cosθ
0 = AB cosθ ⇒ cosθ = 0 = cos90°
θ = 90°

Q.5(A): If the magnitude of the scalar product of two vectors equals the magnitude of their vector product, find the angle between them.

A⃗ · B⃗ = |A⃗ × B⃗|
AB cosθ = AB sinθ
sinθ/cosθ = 1 ⇒ tanθ = tan45°
θ = 45°

Q.5(B): The dot product of two vectors having magnitudes 3 and 4 is 6. What is the angle between them?

Given: A = 3, B = 4 and A⃗ · B⃗ = 6

A⃗ · B⃗ = AB cosθ
6 = 3 × 4 cosθ
cosθ = 6/12 = 1/2 = cos60°
θ = 60°

Properties of Unit Vectors

Let î, ĵ and k̂ be the unit vectors along the x-axis, y-axis and z-axis respectively.

Mutually perpendicular unit vectors along x y and z axes x (î) y (ĵ) z (k̂) 90° 90° 90°
Unit vectors î, ĵ and k̂ along mutually perpendicular coordinate axes.

(a) Scalar Products

î · î = ĵ · ĵ = k̂ · k̂ = 1
î · ĵ = ĵ · k̂ = k̂ · î = 0

(b) Vector Products

î × î = ĵ × ĵ = k̂ × k̂ = 0
î × ĵ = k̂,   ĵ × k̂ = î,   k̂ × î = ĵ
ĵ × î = −k̂,   k̂ × ĵ = −î,   î × k̂ = −ĵ

Solved Questions on Products of Vectors

Q.6(A)

If î, ĵ and k̂ are unit vectors along x, y and z axes respectively, find î · (ĵ × k̂).

ĵ × k̂ = î
î · (ĵ × k̂) = î · î = cos0° = 1
î · (ĵ × k̂) = 1

Q.6(B)

What does A⃗ · A⃗, the scalar product of a vector with itself, give? What about A⃗ × A⃗, the vector product of a vector with itself?

A⃗ · A⃗ = AA cos0° = A2

Thus, the scalar product of a vector with itself gives the square of its magnitude.

A⃗ × A⃗ = AA sin0° n̂ = 0

Thus, the vector product of a vector with itself gives a null vector (or zero).

Q.6(C): Power from Force and Displacement

A force (in newton) expressed in vector notation as F⃗ = 4î + 7ĵ − 3k̂ is applied on a body and produces a displacement (in meter) D⃗ = 3î − 2ĵ − 5k̂ in 4 seconds. Estimate the power.

Given:

F⃗ = 4î + 7ĵ − 3k̂
D⃗ = 3î − 2ĵ − 5k̂
W = F⃗ · D⃗
= (4î + 7ĵ − 3k̂) · (3î − 2ĵ − 5k̂)
= 12 − 14 + 15 = 13 J
P = W/t = 13/4
P = 3.25 W
Note:

If A⃗ = x1î + y1ĵ + z1k̂ and B⃗ = x2î + y2ĵ + z2k̂, then:

A⃗ · B⃗ = x1x2 + y1y2 + z1z2

If L⃗ = xî + yĵ + zk̂, then:

|L⃗| = √(x2 + y2 + z2)

Numericals and Short Questions

Q.1

The angle between two vectors A⃗ and B⃗ is θ. Find the magnitude and direction of A⃗ × B⃗ and A⃗ · B⃗.

The magnitude of A⃗ × B⃗ is AB sinθ and its direction is n̂, the unit vector perpendicular to the plane containing A⃗ and B⃗.

The scalar product A⃗ · B⃗ has value AB cosθ and has no direction.

Q.2

Given two vectors A⃗ = 4.00î + 3.00ĵ and B⃗ = 5.00î − 2.00ĵ, find the magnitude of each vector.

|A⃗| = √(42 + 32) = 5 units
|B⃗| = √(52 + (−2)2) = √29 = 5.38 units

Q.3

Under what condition is the cross product of two vectors equal to zero?

When two vectors are parallel to each other (θ = 0°), their cross product is equal to zero.

Q.4

If A⃗ and B⃗ are non-zero vectors, is it possible for A⃗ × B⃗ and A⃗ · B⃗ both to be zero? Explain.

The scalar product is zero when the vectors are perpendicular to each other, while the vector product is zero when they are parallel. For two non-zero vectors, they cannot be perpendicular and parallel at the same time. Therefore, it is not possible for both A⃗ × B⃗ and A⃗ · B⃗ to be zero simultaneously.

Q.5

Calculate the angle between a 2 dyne force and a 3 dyne force so that their sum is 4 dyne.

Let A = 2, B = 3, R = 4 and the angle between A⃗ and B⃗ be θ.

R = √(A2 + B2 + 2AB cosθ)
42 = 22 + 32 + 2(2)(3)cosθ
16 = 4 + 9 + 12cosθ
12cosθ = 3 ⇒ cosθ = 1/4
θ = cos−1(1/4)
θ = 75.52°

Long Questions

Q.1

Two forces of 30 N and 40 N are inclined to each other at an angle of 60°. What is their resultant? What will be the resultant if the forces are inclined at right angle to each other?

Given: A = 30 N, B = 40 N.

Case I: θ = 60°

R = √(A2 + B2 + 2AB cosθ)
= √[(30)2 + (40)2 + 2(30)(40)cos60°]
= √[900 + 1600 + 2400 × 1/2]
= √3700
R = 60.82 N

Direction:

α = tan−1[(B sinθ)/(A + B cosθ)]
= tan−1[(40 sin60°)/(30 + 40 cos60°)]
= tan−1[(20√3)/50] = tan−1(2√3/5)
α = 34.71°

Case II: θ = 90°

R = √[(30)2 + (40)2 + 2(30)(40)cos90°]
R = √2500 = 50 N
α = tan−1[(40 sin90°)/(30 + 40 cos90°)]
= tan−1(40/30) = tan−1(4/3)
α = 53.13°

Q.2

At what angle do the two forces (A⃗ + B⃗) and (A⃗ − B⃗) act so that their resultant is √(3A2 + B2)?

Given: R = √(3A2 + B2) and the two forces are A⃗ + B⃗ and A⃗ − B⃗. Let the angle between these two forces be θ.

√(3A2 + B2) = √[(A⃗ + B⃗)2 + (A⃗ − B⃗)2 + 2(A⃗ + B⃗)(A⃗ − B⃗)cosθ]

Squaring both sides and following the source derivation:

3A2 + B2 = A2 + 2A⃗·B⃗ + B2 + A2 − 2A⃗·B⃗ + B2 + 2cosθ(A2 − B2)
A2 − B2 = 2cosθ(A2 − B2)
2cosθ = 1 ⇒ cosθ = 1/2 = cos60°
θ = 60°

Q.3

Find the scalar product of the two vectors A⃗ = 4.00î + 3.00ĵ and B⃗ = 5.00î − 2.00ĵ. Also find the angle between them.

A = √(42 + 32) = 5 units
B = √(52 + (−2)2) = √29 units
A⃗ · B⃗ = (4î + 3ĵ) · (5î − 2ĵ)
= 20 − 6 = 14
A⃗ · B⃗ = 14
A⃗ · B⃗ = AB cosθ
14 = 5√29 cosθ
cosθ = 14/(5√29)
θ = cos−1[14/(5√29)]
θ = 58.67°

Difference Between Scalar Product and Vector Product

Scalar (Dot) Product Vector Product (Cross Product)
The product is said to be scalar if two vectors multiplied together give a scalar quantity. The product is said to be vector if two vectors multiplied together give a vector quantity.
It is denoted by A⃗ · B⃗. It is denoted by A⃗ × B⃗.
A⃗ · B⃗ = AB cosθ A⃗ × B⃗ = AB sinθ n̂
It is commutative. It is anti-commutative.
The scalar product of a vector with itself gives the square of its magnitude. The vector product of a vector with itself gives a null vector.
When two vectors are parallel, the scalar product is AB. When two vectors are parallel, the vector product is zero.
When two vectors are perpendicular to each other, the dot product is zero. When two vectors are perpendicular to each other, the vector product is AB n̂.
When two vectors are antiparallel to each other, the scalar product is −AB. When two vectors are antiparallel to each other, the vector product is 0.
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