Vectors: Complete Chapter Notes For Class 11 NEET by PW

Magnitude, direction, components and vector operations can make Vectors difficult to revise when the concepts are studied separately. NEET questions can require you to calculate a resultant, resolve a vector into components or identify whether a dot product or cross product should be used.

The PW Vectors Complete Chapter One-Shot Revision Video For Class 11 NEET brings these concepts together for quick revision. These notes cover the important vector laws, types, formulas, components and products to help you revise the chapter before practising NEET questions.

Physical Quantities

Physical quantities are measurable quantities used to describe physical phenomena and express the laws of physics. They are represented using a numerical value and an appropriate unit.

Physical quantities are broadly divided into:

1. Scalar Physical Quantities

2. Vector Physical Quantities

Scalar Physical Quantities

A scalar physical quantity has magnitude only and does not require a direction for its complete description.

Examples include:

• Mass

• Density

• Volume

• Speed

• Distance

Scalars are added and subtracted using ordinary algebraic rules:

2 kg + 3 kg = 5 kg

2 L - 0.5 L = 1.5 L

Magnitude

The magnitude of a physical quantity refers to its numerical value expressed with its unit. For example, the magnitude of a mass of 5 kg is 5 kg.

Vector Physical Quantities

A vector physical quantity has both magnitude and direction and follows the rules of vector addition.

Examples include:

• Displacement

• Velocity

• Acceleration

• Force

• Momentum

• Torque

For example, force is a vector because its effect depends not only on how large the force is but also on the direction in which it acts.

Scalar And Vector Comparison

Direction alone does not make a quantity a vector. A vector must have both magnitude and direction and must obey vector addition laws.

Memory Tip: Current is treated as a scalar quantity in elementary physics because currents are added algebraically rather than according to vector addition laws. NCERT also notes this treatment explicitly.

Symbolic And Graphical Representation Of Vectors

A vector can be represented using a bold symbol, such as F, v or a. This notation avoids the arrow-over-symbol formatting that can create rendering issues in Google Docs.

Graphically, a vector is represented by a directed line segment:

• Tail: Starting point

• Head: Pointed end

• Direction: From the tail towards the head

• Magnitude: Represented by the length of the arrow, according to the chosen scale

Magnitude of F = |F|

A vector can be translated parallel to itself without changing its magnitude or direction. Therefore, a free vector remains unchanged when shifted parallel to itself.

A vector returns to its original direction after a complete rotation of:

2nπ

where n is an integer.

Special Addition And Subtraction Cases

For vectors along the same straight line:

• Vectors in the same direction are added.

• Vectors in opposite directions are subtracted.

• The resultant of two opposite vectors acts in the direction of the vector with greater magnitude.

For example:

3 N + 4 N = 7 N

10 N - 3 N = 7 N

These are one-dimensional cases of vector addition and should not be confused with ordinary scalar addition in situations where vectors act at an angle.

Types Of Vectors

Parallel Vectors

Vectors whose directions are the same or opposite along parallel lines are called parallel vectors. Their magnitudes may be equal or unequal.

Equal Vectors

Two vectors are equal when they have the same magnitude and the same direction.

All equal vectors are parallel, but all parallel vectors are not necessarily equal.

Antiparallel Vectors

Vectors that act in opposite directions along parallel lines are called antiparallel vectors. Their magnitudes may be equal or unequal.

Opposite Vectors

Opposite vectors have equal magnitudes and opposite directions.

All opposite vectors are antiparallel, but all antiparallel vectors are not necessarily opposite.

Perpendicular Vectors

Two vectors are perpendicular or orthogonal when the angle between them is:

90°

Unit Vectors

A unit vector has a magnitude of one:

|A| = 1

The standard unit vectors along the x, y and z axes are represented by:

i, j, k

Their magnitudes are:

|i| = |j| = |k| = 1

Memory Tip: Unit vectors indicate direction while having a magnitude of 1.

Concurrent, Collinear And Coplanar Vectors

• Concurrent vectors: Their lines of action pass through one common point.

• Collinear vectors: Their lines of action lie along the same straight line.

• Coplanar vectors: They lie in the same plane.

Any two vectors are always coplanar. For three vectors, the scalar triple product can be used to test coplanarity:

(A × B) · C = 0

Null Vector

A null or zero vector has zero magnitude:

|0| = 0

Its direction is undefined.

For example, equal and opposite vectors have a zero resultant:

A + (-A) = 0

Axial Vectors

An axial vector is associated with rotational motion and is directed along the axis of rotation according to the right-hand rule.

Examples include:

• Torque

• Angular velocity

• Angular acceleration

• Angular momentum

Angular displacement can also be represented as an axial vector for rotational motion.

Vector Addition

Vector addition can be performed using geometrical laws or by resolving vectors into components.

Triangle Law

According to the triangle law, two vectors are arranged head to tail. The resultant is drawn from the tail of the first vector to the head of the second vector.

A + B = R

Memory Tip: Place two vectors head to tail, then join the first tail to the final head to obtain the resultant.

Polygon Law

The polygon law applies when two or more vectors are placed consecutively, head to tail.

The resultant joins the tail of the first vector to the head of the last vector.

For a closed vector polygon:

A + B + C = 0

Therefore, the resultant of vectors forming a closed loop is zero.

Parallelogram Law

When two vectors are placed tail to tail, they form adjacent sides of a parallelogram. The diagonal from their common tail gives the resultant.

If the magnitudes are a and b, and the angle between them is θ, then:

R = √(a² + b² + 2ab cos θ)

The resultant lies between the two vectors and is closer in direction to the vector with the greater magnitude, when their magnitudes are unequal.

Important Special Cases

For two vectors of magnitudes a and b, the resultant can have any value within:

|a - b| ≤ R ≤ a + b

The maximum resultant occurs when the vectors are parallel and in the same direction. The minimum occurs when they are antiparallel.

Vector Subtraction

Vector subtraction is treated as the addition of a negative vector:

A - B = A + (-B)

The vector -B has the same magnitude as B but points in the opposite direction.

The magnitude of the difference between two vectors is:

|A - B| = √(a² + b² - 2ab cos θ)

Vector subtraction is not commutative:

A - B ≠ B - A

For two vectors of equal magnitude F₀:

|A + B| = 2F₀ cos(θ/2)

|A - B| = 2F₀ sin(θ/2)

Components Of A Vector

A vector can be resolved into rectangular components along the coordinate axes.

If a vector F makes an angle θ with the positive x-axis:

Fₓ = F cos θ

Fᵧ = F sin θ

Memory Tip: When the angle is measured from the x-axis, the x-component uses cosine and the y-component uses sine.

The Cartesian form of a vector is:

A = Aₓ i + Aᵧ j + A_z k

Its magnitude is:

|A| = √(Aₓ² + Aᵧ² + A_z²)

Vector addition and subtraction can be performed by adding or subtracting the corresponding components.

The unit vector along A is:

 = A / |A|

Directions

Position And Displacement Vectors

The position vector of a point P(x, y, z) measured from the origin is:

r = x i + y j + z k

Displacement is the difference between the final and initial position vectors:

AB = rB - rA

If:

A(x₁, y₁, z₁)

and

B(x₂, y₂, z₂)

then:

AB = (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k

Thus, displacement depends only on the initial and final positions, not on the path followed between them.

Dot Product Or Scalar Product

The dot product, also called the scalar product, of two vectors is:

A · B = |A||B| cos θ

The result of a dot product is a scalar.

Its sign depends on the angle between the vectors:

• Acute angle: Positive dot product

• 90°: Zero dot product

• Obtuse angle: Negative dot product

In Cartesian form:

A · B = AₓBₓ + AᵧBᵧ + A_zB_z

Important applications include:

W = F · d

P = F · v

The angle between two non-zero vectors is:

θ = cos⁻¹[(A · B) / (|A||B|)]

Scalar Projection

The scalar projection of A along B is:

(A · B) / |B|

It represents the component of A along the direction of B.

NCERT also describes the dot product in terms of the magnitude of one vector and the projection of the other along it.

Cross Product Or Vector Product

The cross product, also called the vector product, is defined as:

|A × B| = |A||B| sin θ

The result is a vector perpendicular to the plane containing A and B.

Important cases:

• Parallel vectors: Cross product is zero.

• Antiparallel vectors: Cross product is zero.

• Perpendicular vectors: Cross product has maximum magnitude.

The direction is determined using the right-hand rule.

The cross product is anti-commutative:

A × B = -(B × A)

For the standard unit vectors:

i × j = k

j × k = i

k × i = j

Reversing the order changes the sign:

j × i = -k

k × j = -i

i × k = -j

Cross Product In Component Form

For:

A = Aₓi + Aᵧj + A_zk

B = Bₓi + Bᵧj + B_zk

The cross product can be written as:

A × B =

| i j k |
| Aₓ Aᵧ A_z |
| Bₓ Bᵧ B_z |

When using Google Docs, this determinant can also be expanded directly into components if the determinant formatting does not render correctly.

Applications Of Cross Product

Torque is given by:

τ = r × F

Angular momentum is:

L = r × p

The area of the parallelogram formed by two vectors is:

A(parallelogram) = |A × B|

The area of the triangle formed by the same two vectors is:

A(triangle) = 1/2 |A × B|

Vectors form the foundation for several Class 11 Physics topics, including motion, force, work, torque and angular momentum. Revise vector addition, components, resultants, dot product and cross product carefully, and practise numerical questions to strengthen their application. The PW Vectors Complete Chapter One-Shot Revision Video For Class 11 NEET can help you revise the key concepts and formulas in one session. 

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