CBSE Class 11 Physics Vectors Notes, MCQs, Questions & Answers | NEET Preparation

 - Dr.Sanjaykumar Pawar  

Vectors Class 11 Physics Notes, MCQs, Assertion Reason, Case Study & CBSE Questions

Complete Class 11 Physics Vectors Notes with formulas, diagrams, MCQs, assertion-reason, case studies and CBSE exam questions.

Internal Links

• Class 11 Physics Units and Measurements
• Motion in a Straight Line Notes
• Motion in a Plane
• Projectile Motion
• Laws of Motion
• Work, Energy and Power
• System of Particles
• Circular Motion
• Kinematics Formula Sheet
• Physics Formula Handbook
• NEET Physics Notes
• CBSE Class 11 Physics MCQs
• Class 11 Physics Previous Year Questions
• Class 11 Physics Sample Papers
• NCERT Solutions for Class 11 Physics
• Important Physics Derivations
• Physics Practical Experiments
• Physics Revision Notes
• Physics Chapter-wise Question Bank
• CBSE Class 11 Study Material

NEET Physics Chapter : Vectors

Vectors are one of the most important topics in NEET Physics. Almost every chapter uses vectors. Therefore understanding vectors properly makes Mechanics very easy.

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1. Equality of Vectors

Two vectors are called equal if

• Magnitude is same.
• Direction is same.

Position does NOT matter. Even if vectors are shifted parallel, they are still equal.

Equal vectors ⇒ Same Magnitude + Same Direction

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2. Addition of Vectors

Vector addition means combining two vectors to get one resultant vector.

Triangle Law

Place the tail of second vector at the head of first vector. Join the starting point to the final point. That gives resultant.

Remember: Head to Tail Rule

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3. Parallelogram Law

If two vectors start from the same point, complete a parallelogram. Diagonal gives resultant vector.

Resultant = Diagonal of Parallelogram

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4. Magnitude of Resultant

Suppose

First Vector = a

Second Vector = b

Angle between them = θ

R = √(a² + b² + 2ab cosθ)

This is one of the MOST IMPORTANT formulas for NEET. Learn it perfectly.

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5. Direction of Resultant

tanα = (b sinθ)/(a + b cosθ)

α = angle made by resultant with first vector.

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6. Special Cases

Angle	Magnitude
0°	a+b
180°	|a-b|
90°	√(a²+b²)

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7. Example

Question: Two vectors have equal magnitude A. Angle between them is θ. Find resultant.

Solution

R = √(A²+A²+2A²cosθ)

= √(2A²(1+cosθ))

Using 1+cosθ=2cos²(θ/2)

R = 2A cos(θ/2)

Resultant = 2A cos(θ/2)

Direction:

α = θ/2

The resultant bisects the angle between two equal vectors.

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8. Memory Tricks

✔ Triangle Rule → Head to Tail

✔ Parallelogram Rule → Diagonal

✔ Equal Vectors → Same Magnitude + Same Direction

✔ 90° → Pythagoras

✔ 180° → Subtraction

✔ 0° → Addition

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9. NEET Important Points

• Magnitude is always positive.
• Direction decides vector.
• Vectors obey triangle law.
• Resultant depends on angle.
• Equal vectors can have different positions.
• Parallelogram law is frequently asked in NEET.

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10. Practice Questions

1. Define equal vectors.
2. State triangle law.
3. State parallelogram law.
4. Write magnitude formula.
5. Write direction formula.
6. Find resultant when angle is 90°.
7. Find resultant when angle is 180°.
8. Two vectors 10 N each make 60°. Find resultant.
9. Two vectors 5 N each make 120°. Find resultant.
10. Why does the resultant bisect equal vectors?

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Summary

• Equal vectors → Same magnitude + same direction
• Triangle Law → Head to Tail
• Parallelogram Law → Diagonal
• Magnitude → √(a²+b²+2abcosθ)
• Direction → tanα=(bsinθ)/(a+bcosθ)
• Equal vectors → Resultant = 2Acos(θ/2)
• Direction = θ/2

CBSE Class 11 Physics

Chapter : Vectors Question Bank

1. Multiple Choice Questions (MCQs)

1. A vector quantity has

1. Only magnitude
2. Only direction
3. Magnitude and direction
4. None

Answer: C

2. Equal vectors have

1. Equal magnitude only
2. Equal direction only
3. Equal magnitude and direction
4. Different directions

Answer: C

3. The diagonal of a parallelogram represents

1. Difference of vectors
2. Resultant vector
3. Unit vector
4. Zero vector

Answer: B

2. Very Short Answer Questions (1 Mark)

Q1. Define a vector.

A quantity having both magnitude and direction is called a vector.

Q2. Give one example of a vector.

Velocity.

Q3. What is a zero vector?

A vector whose magnitude is zero.

3. Short Answer Questions (2-3 Marks)

Q1. Define equal vectors.

Two vectors having equal magnitude and same direction are called equal vectors.

Q2. State the triangle law of vector addition.

If two vectors are represented by two sides of a triangle taken in order, the third side taken in opposite order represents the resultant.

4. Long Answer Questions (5 Marks)

Q1. Explain the parallelogram law of vector addition with diagram.

If two vectors acting simultaneously are represented by two adjacent sides of a parallelogram, then the diagonal passing through the common point represents the resultant vector. Magnitude: R = √(A² + B² + 2AB cosθ) Direction: tanα = (B sinθ)/(A + B cosθ)

5. Assertion and Reason

Assertion: Equal vectors may have different positions.

Reason: A vector depends only on magnitude and direction.

Answer: Both Assertion and Reason are true, and Reason correctly explains Assertion.

Assertion: Scalar quantities have direction.

Reason: Scalars possess only magnitude.

Answer: Assertion is false. Reason is true.

6. Fill in the Blanks

Question	Answer
A vector has ______ and ______.	Magnitude, Direction
The diagonal of a parallelogram gives the ______.	Resultant
A quantity having only magnitude is called ______.	Scalar
The SI unit of displacement is ______.	metre

7. Match the Columns

Column A	Column B
Velocity	Vector
Mass	Scalar
Acceleration	Vector
Time	Scalar

Answers Velocity → Vector Mass → Scalar Acceleration → Vector Time → Scalar

8. Statement Based Questions

Statement I: The resultant of two equal vectors bisects the angle between them.

Statement II: The magnitudes of both vectors are equal.

Both statements are true. Statement II explains Statement I.

9. Case Study Questions

Rahul pushes a box with force 20 N towards east. Aman pushes the same box with force 20 N making an angle of 60°. Answer the following.

1. Which law is used?
2. Write the magnitude formula.
3. If angle becomes 180°, what happens?

1. Parallelogram law.
2. R = √(A²+B²+2ABcosθ)
3. Resultant = |A−B|

10. Numerical Questions

Two vectors of magnitude 5 N each make an angle of 60°. Find the resultant.

R = √(25+25+50×0.5) = √75 = 8.66 N

11. HOTS Questions

Why can two vectors be equal even if they are drawn at different places?

Because a vector depends only on magnitude and direction, not on position.

12. Competency Based Questions

A boat is moving across a river. Which physical quantities should be treated as vectors?

Velocity, displacement and acceleration.

13. One Word Questions

Question	Answer
Quantity having direction	Vector
Quantity having only magnitude	Scalar
Magnitude zero vector	Zero Vector
Vector of magnitude one	Unit Vector

14. Important CBSE Questions

1. Define vector.
2. State triangle law.
3. State parallelogram law.
4. Define equal vectors.
5. What is a unit vector?
6. What is a null vector?
7. Derive the magnitude formula.
8. Derive the direction formula.
9. Differentiate scalar and vector.
10. Give five examples each of scalars and vectors.

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