Scalar Product (Dot Product) Class 11 Physics Notes for NEET

 Scalar Product Explained Easily | NEET Physics Vector Notes

- Dr.Sanjaykumar Pawar 

Scalar Product (Dot Product) of two vectors showing angle, projection and formula A·B = AB cosθ.

INTERNAL LINK SUGGESTIONS

Introduction to Vectors in Physics

Types of Physical Quantities: Scalars and Vectors

Vector Addition and Subtraction

Resolution of Vectors

Unit Vectors Explained

Position Vector Notes

Cross Product (Vector Product)

Motion in a Plane Notes

Projectile Motion Complete Guide

Laws of Motion Notes

Work, Energy and Power

Rotational Motion Basics

Important Vector Formulas for NEET

NCERT Class 11 Physics Chapter-wise Notes

NEET Physics Formula Handbook

The Scalar Product (Dot Product) - NEET Notes

1. Introduction

• Physical quantities such as displacement, velocity, acceleration and force are vectors.
• A vector has both magnitude and direction.
• We already know how vectors are added and subtracted.
• Now we study how vectors are multiplied.

Types of Vector Multiplication

1. Scalar Product (Dot Product) → Produces a scalar quantity.
2. Vector Product (Cross Product) → Produces a new vector.

In this chapter we study the Scalar Product (Dot Product).

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2. Definition of Scalar Product

The scalar product or dot product of two vectors A and B is written as:

A · B = AB cosθ

where:

• A = Magnitude of vector A
• B = Magnitude of vector B
• θ = Angle between vectors A and B

Since A, B and cosθ are scalars, the dot product is also a scalar quantity.

The vectors A and B have directions, but their scalar product has no direction.

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3. Geometrical Meaning of Dot Product

A · B = A(B cosθ)

B cosθ is the projection (component) of vector B along vector A.

Therefore:

• Dot Product = Magnitude of A × Component of B along A

A · B = B(A cosθ)

A cosθ is the projection of vector A along vector B.

• Dot Product = Magnitude of B × Component of A along B

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4. Special Cases of Dot Product

Case 1: θ = 0°

A · B = AB cos0°
A · B = AB

Maximum positive value.

Case 2: θ = 90°

A · B = AB cos90°
A · B = 0

Vectors are perpendicular.

Case 3: θ = 180°

A · B = AB cos180°
A · B = -AB

Maximum negative value.

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5. Properties of Scalar Product

A. Commutative Law

A · B = B · A

Changing the order does not change the answer.

B. Distributive Law

A · (B + C) = A · B + A · C

C. Scalar Multiplication

A · (λB) = λ(A · B)

where λ is a real number.

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6. Dot Product of Unit Vectors

The unit vectors are:

î , ĵ , k̂

• î → x-axis direction
• ĵ → y-axis direction
• k̂ → z-axis direction

Same Unit Vectors

î · î = 1
ĵ · ĵ = 1
k̂ · k̂ = 1

Same unit vectors → Answer = 1

Different Unit Vectors

î · ĵ = 0
ĵ · k̂ = 0
k̂ · î = 0

Different unit vectors → Answer = 0

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7. Cartesian Form of Vectors

A = Ax î + Ay ĵ + Az k̂

B = Bx î + By ĵ + Bz k̂

Then their scalar product is:

A · B = AxBx + AyBy + AzBz

Very Important Formula for NEET

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8. Dot Product of a Vector with Itself

A · A = Ax² + Ay² + Az²

Also,

A · A = |A||A| cos0°
A · A = A²

Therefore,

A² = Ax² + Ay² + Az²

Magnitude of vector A:

|A| = √(Ax² + Ay² + Az²)

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9. Condition for Perpendicular Vectors

If two vectors are perpendicular:

θ = 90°

Since cos90° = 0:

A · B = 0

If A · B = 0, then vectors A and B are perpendicular (provided neither vector is zero).

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NEET Quick Revision

Definition

A · B = AB cosθ

Component Form

A · B = AxBx + AyBy + AzBz

Unit Vector Results

î · î = 1
ĵ · ĵ = 1
k̂ · k̂ = 1

î · ĵ = 0
ĵ · k̂ = 0
k̂ · î = 0

Magnitude Formula

|A| = √(Ax² + Ay² + Az²)

Perpendicular Vectors

A · B = 0

Special Angles

Angle (θ)	Dot Product
0°	AB
90°	0
180°	-AB

Remember:

Dot Product = Magnitude × Magnitude × cos(angle)

The dot product tells how much one vector acts in the direction of another vector.

CBSE Class 11 Physics

Chapter: Scalar Product (Dot Product)

A. Multiple Choice Questions (MCQs)

1. The scalar product of two vectors is always a:

(a) Vector    (b) Scalar    (c) Tensor    (d) Matrix

Answer: (b) Scalar

2. The dot product of two perpendicular vectors is:

(a) 1    (b) -1    (c) 0    (d) Infinity

Answer: (c) 0

3. The scalar product formula is:

(a) A × B    (b) A + B    (c) AB cosθ    (d) A − B

Answer: (c) AB cosθ

4. The value of î · ĵ is:

(a) 1    (b) 0    (c) -1    (d) 2

Answer: (b) 0

5. A · A equals:

(a) A²    (b) A    (c) 0    (d) 1

Answer: (a) A²

B. Very Short Answer Questions (1 Mark)

1. Define scalar product.

The scalar product of vectors A and B is defined as: A · B = AB cosθ

2. What is î · î ?

1

3. What is ĵ · k̂ ?

0

4. What is the angle between vectors having zero dot product?

90°

5. Is dot product scalar or vector?

Scalar

C. Short Answer Questions (2–3 Marks)

1. State any two properties of scalar product.

1. A · B = B · A (Commutative Law)
2. A · (B + C) = A · B + A · C (Distributive Law)

2. Find the dot product of A = 2î + 3ĵ and B = 4î + 5ĵ.

A · B = (2×4) + (3×5)
= 8 + 15
= 23

3. Why is scalar product called a scalar quantity?

Because the result of the multiplication has magnitude only and no direction.

D. Long Answer Questions (5 Marks)

1. Define scalar product and explain its geometrical significance.

The scalar product of vectors A and B is:A · B = AB cosθ where θ is the angle between the vectors. Geometrically, B cosθ is the projection of B along A. Therefore, A · B = A(B cosθ) Similarly, A cosθ is the projection of A along B. Thus, scalar product represents the product of the magnitude of one vector and the component of the other vector along it.

2. Derive the Cartesian form of scalar product.

A = Ax î + Ay ĵ + Az k̂ B = Bx î + By ĵ + Bz k̂ Using: î·î = 1 ĵ·ĵ = 1 k̂·k̂ = 1 î·ĵ = ĵ·k̂ = k̂·î = 0 Therefore, A · B = AxBx + AyBy + AzBz Hence proved.

E. Assertion and Reason Questions

Assertion (A): The dot product of two perpendicular vectors is zero.
Reason (R): cos 90° = 0.

Answer: Both A and R are true and R is the correct explanation.

Assertion (A): î · ĵ = 1
Reason (R): î and ĵ are perpendicular.

Answer: Assertion is false but Reason is true.

F. Fill in the Blanks

1. Scalar product of vectors A and B is __________.
2. Dot product of perpendicular vectors is __________.
3. î · î = __________.
4. ĵ · k̂ = __________.
5. A · A = __________.

1. AB cosθ
2. Zero
3. 1
4. 0
5. A²

G. True or False

Statement	Answer
Dot product gives a vector quantity.	False
A · B = B · A	True
î · ĵ = 1	False
Dot product can be negative.	True
A · A is always positive.	True

H. Match the Columns

Column A	Column B
î·î	1
î·ĵ	0
A·A	A²
Perpendicular vectors	A·B = 0

Answers:
1 → 1
2 → 0
3 → A²
4 → A·B = 0

I. Case Study Questions

Two vectors A and B have magnitudes 5 and 10 units respectively. The angle between them is 60°.

1. What is the value of cos 60°?

1/2

2. Calculate A · B.

A · B = AB cosθ
= 5 × 10 × 1/2
= 25

3. Is the result scalar or vector?

Scalar

4. What will be the dot product if θ = 90°?

0

5. What will be the dot product if θ = 180°?

-50

J. Competency-Based Questions

1. A student claims that if A·B = 0, then vectors are perpendicular. Is the statement correct?

Yes. For non-zero vectors,A·B = AB cosθ = 0 Therefore cosθ = 0 Hence θ = 90°. The vectors are perpendicular.

2. Why is scalar product useful in physics?

It is used in calculating work done, power, projections of vectors and determining angles between vectors.

THE SCALAR PRODUCT (DOT PRODUCT)

THE SCALAR PRODUCT (DOT PRODUCT)
│
├── Introduction
│   │
│   ├── Vectors have
│   │   ├── Magnitude
│   │   └── Direction
│   │
│   ├── Examples
│   │   ├── Displacement
│   │   ├── Velocity
│   │   ├── Acceleration
│   │   └── Force
│   │
│   └── Vector Multiplication
│       ├── Scalar Product (Dot Product)
│       └── Vector Product (Cross Product)
│
├── Definition
│   │
│   ├── A · B = AB cosθ
│   │
│   ├── A = Magnitude of Vector A
│   ├── B = Magnitude of Vector B
│   ├── θ = Angle Between Vectors
│   │
│   └── Result
│       └── Scalar Quantity
│
├── Geometrical Meaning
│   │
│   ├── A · B = A(B cosθ)
│   │   └── B cosθ = Component of B along A
│   │
│   ├── A · B = B(A cosθ)
│   │   └── A cosθ = Component of A along B
│   │
│   └── Dot Product
│       └── Measures Projection
│
├── Special Cases
│   │
│   ├── θ = 0°
│   │   ├── cos0° = 1
│   │   └── A · B = AB
│   │
│   ├── θ = 90°
│   │   ├── cos90° = 0
│   │   └── A · B = 0
│   │
│   └── θ = 180°
│       ├── cos180° = -1
│       └── A · B = -AB
│
├── Properties
│   │
│   ├── Commutative Law
│   │   └── A · B = B · A
│   │
│   ├── Distributive Law
│   │   └── A · (B + C)
│   │       = A · B + A · C
│   │
│   └── Scalar Multiplication
│       └── A · (λB)
│           = λ(A · B)
│
├── Unit Vectors
│   │
│   ├── î → x-axis
│   ├── ĵ → y-axis
│   └── k̂ → z-axis
│
├── Dot Product of Same Unit Vectors
│   │
│   ├── î · î = 1
│   ├── ĵ · ĵ = 1
│   └── k̂ · k̂ = 1
│
├── Dot Product of Different Unit Vectors
│   │
│   ├── î · ĵ = 0
│   ├── ĵ · k̂ = 0
│   └── k̂ · î = 0
│
├── Cartesian Form
│   │
│   ├── A = Ax î + Ay ĵ + Az k̂
│   │
│   ├── B = Bx î + By ĵ + Bz k̂
│   │
│   └── A · B
│       └── AxBx + AyBy + AzBz
│
├── Vector with Itself
│   │
│   ├── A · A
│   │   └── Ax² + Ay² + Az²
│   │
│   ├── A · A = A²
│   │
│   └── Magnitude
│       └── |A|
│           = √(Ax² + Ay² + Az²)
│
├── Perpendicular Vectors
│   │
│   ├── θ = 90°
│   ├── cos90° = 0
│   └── A · B = 0
│
└── NEET Quick Revision
    │
    ├── Formula
    │   └── A · B = AB cosθ
    │
    ├── Component Form
    │   └── AxBx + AyBy + AzBz
    │
    ├── Same Unit Vectors
    │   └── Answer = 1
    │
    ├── Different Unit Vectors
    │   └── Answer = 0
    │
    ├── Magnitude
    │   └── √(Ax² + Ay² + Az²)
    │
    ├── Perpendicular
    │   └── A · B = 0
    │
    └── Remember
        └── Dot Product
            = Magnitude × Magnitude × cosθ