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.

---

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

---

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

---

3. Parallelogram Law

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

Resultant = Diagonal of Parallelogram

---

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.

---

5. Direction of Resultant

tanα = (b sinθ)/(a + b cosθ)

α = angle made by resultant with first vector.

---

6. Special Cases

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

---

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.

---

8. Memory Tricks

✔ Triangle Rule → Head to Tail

✔ Parallelogram Rule → Diagonal

✔ Equal Vectors → Same Magnitude + Same Direction

✔ 90° → Pythagoras

✔ 180° → Subtraction

✔ 0° → Addition

---

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.

---

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?

---

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.

CBSE Class 11 Physics Vector Addition Notes, MCQs & Questions

 RESULTANT OF TWO VECTORS

│

├── Given

│   ├── Vector A

│   ├── Vector B

│   └── Angle between vectors = θ

│

├── Vector Addition

│   ├── Use Parallelogram Law

│   ├── Resultant vector = R

│   └── R = A + B

│

├── Components of Vector B

│   ├── Horizontal Component

│   │   └── B cos θ

│   │

│   └── Vertical Component

│       └── B sin θ

│

├── Geometry Relations

│   ├── ON = A + B cos θ

│   └── SN = B sin θ

│

├── Pythagoras Theorem

│   ├── R² = ON² + SN²

│   ├── R² = (A + B cos θ)² + (B sin θ)²

│   └── Simplified:

│       └── R² = A² + B² + 2AB cos θ

│

├── Magnitude Formula

│   └── R = √(A² + B² + 2AB cos θ)

│

├── Direction of Resultant

│   │

│   ├── Using Sine Relation

│   │   ├── R sin α = B sin θ

│   │   └── sin α = (B sin θ)/R

│   │

│   └── Using Tangent Relation

│       └── tan α = (B sin θ)/(A + B cos θ)

│

├── Important Laws

│   ├── Law of Cosines

│   │   └── R² = A² + B² + 2AB cos θ

│   │

│   └── Law of Sines

│       └── R/sin θ = A/sin β = B/sin α

│

├── Special Cases

│   │

│   ├── θ = 0°

│   │   └── R = A + B

│   │

│   ├── θ = 180°

│   │   └── R = |A − B|

│   │

│   └── θ = 90°

│       └── R = √(A² + B²)

│

└── NEET Quick Revision

    ├── Resolve vectors into components

    ├── Apply Pythagoras theorem

    ├── Use cosine formula for magnitude

    └── Use tangent formula for direction

Parallelogram law showing resultant of two vectors A and B for Class 11 Physics.

-Dr.Sanjaykumar pawar

Example 3.2 - Resultant of Two Vectors

Example 3.2 Find the magnitude and direction of the resultant of two vectors A and B in terms of their magnitudes and angle θ between them Let two vectors be:

• Vector A
• Vector B
• Angle between them = θ

We have to find:

• Magnitude of resultant vector R
• Direction of resultant vector

---

Step 1: Draw the vectors

Draw vector OP representing vector A.

Draw vector OQ representing vector B.

The angle between the vectors is θ.

Using the parallelogram law of vector addition, diagonal OS gives the resultant vector.

R = A + B

Step 2: Resolve vector B into components

Draw perpendicular SN on OP.

Now vector B has two components:

Horizontal Component

B cos θ

Vertical Component

B sin θ

Step 3: Find ON and SN

From geometry:

ON = OP + PN

But,

OP = A

PN = B cos θ

Therefore,

ON = A + B cos θ

Also,

SN = B sin θ

Step 4: Apply Pythagoras Theorem

In right triangle OSN:

OS² = ON² + SN²

Substituting values:

R² = (A + B cos θ)² + (B sin θ)²

Step 5: Expand the Equation

R² = A² + 2AB cos θ + B² cos² θ + B² sin² θ

Take B² common:

R² = A² + 2AB cos θ + B²(cos² θ + sin² θ)

Using identity:

sin² θ + cos² θ = 1

Therefore,

R² = A² + B² + 2AB cos θ

Final Formula for Magnitude

R = √(A² + B² + 2AB cos θ)

This formula gives the magnitude of the resultant vector.

This is called the Law of Cosines.

---

Direction of Resultant Vector

Let the resultant vector make angle α with vector A.

Step 6: Use Sine Relation

From triangle:

SN = R sin α

Also,

SN = B sin θ

Equating both:

R sin α = B sin θ

Therefore,

sin α = (B sin θ) / R

Step 7: Formula for tan α

From triangle:

tan α = SN / ON

Substitute values:

tan α = (B sin θ) / (A + B cos θ)

---

Important Results for NEET

Magnitude of Resultant

R = √(A² + B² + 2AB cos θ)

Direction of Resultant

tan α = (B sin θ) / (A + B cos θ)

---

Special Cases

1. When θ = 0°

R = A + B

2. When θ = 180°

R = |A - B|

3. When θ = 90°

R = √(A² + B²)

---

Quick Concept Summary

• Resultant vector is found using parallelogram law.
• Resolve vector B into horizontal and vertical components.
• Apply Pythagoras theorem to find magnitude.
• Magnitude formula comes from Law of Cosines.
• Direction formula comes from trigonometric ratios.

CBSE Class 11 Physics

Vector Addition - Important Questions and Answers

1. Multiple Choice Questions (MCQs)

Q1. The magnitude of resultant of two vectors A and B inclined at angle θ is:

A) A + B
B) A - B
C) √(A² + B² + 2AB cosθ)
D) √(A² + B²)

Answer: C) √(A² + B² + 2AB cosθ)

Q2. If two vectors act in the same direction, the resultant is:

A) A - B
B) A + B
C) AB
D) Zero

Answer: B) A + B

Q3. If angle between two vectors is 180°, the resultant is:

A) A + B
B) |A - B|
C) Zero
D) AB

Answer: B) |A - B|

2. Very Short Answer Questions

Q1. What is a resultant vector?

A single vector that represents the combined effect of two or more vectors is called resultant vector.

Q2. Which method is used to add two vectors geometrically?

Parallelogram law of vector addition.

Q3. What is the resultant when two vectors are perpendicular?

R = √(A² + B²)

Q4. At which angle is resultant maximum?

Resultant is maximum when angle is 0°.

3. Short Answer Questions

Q1. State parallelogram law of vector addition.

If two vectors acting simultaneously on a particle are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the parallelogram passing through the common point.

Q2. Write formula for magnitude of resultant vector.

R = √(A² + B² + 2AB cosθ)

Where:

• A and B are magnitudes of vectors
• θ is angle between them

Q3. Write formula for direction of resultant vector.

tanα = (B sinθ) / (A + B cosθ)

4. Long Answer Questions

Q1. Derive formula for magnitude of resultant vector.

Consider two vectors A and B inclined at angle θ.

Using parallelogram law:

R = A + B

Apply Pythagoras theorem:

R² = (A + B cosθ)² + (B sinθ)²

Expanding:

R² = A² + 2AB cosθ + B² cos²θ + B² sin²θ

Using identity:

sin²θ + cos²θ = 1

Therefore:

R² = A² + B² + 2AB cosθ

Hence,

R = √(A² + B² + 2AB cosθ)

Q2. Derive formula for direction of resultant vector.

From the triangle:

tanα = SN / ON

Where:

SN = B sinθ

ON = A + B cosθ

Therefore:

tanα = (B sinθ)/(A + B cosθ)

5. Assertion and Reason Questions

Q1.

Assertion (A): Resultant of two equal and opposite vectors is zero.

Reason (R): Opposite vectors cancel each other.

Both Assertion and Reason are true and Reason is the correct explanation.

Q2.

Assertion (A): Resultant of perpendicular vectors is equal to sum of vectors.

Reason (R): Pythagoras theorem is used for perpendicular vectors.

Assertion is false but Reason is true.

6. Fill in the Blanks

Q1. The diagonal of parallelogram gives the _________ vector.

Resultant

Q2. The formula for resultant vector uses law of _________.

Cosines

Q3. If θ = 0°, resultant is _________.

A + B

7. Case Study Questions

Two students are pulling a box using two ropes. One student applies force A and another applies force B making angle θ between them. The combined effect produces resultant force R.

Q1. Which law is used to find resultant force?

Parallelogram law of vector addition.

Q2. Write formula for resultant force.

R = √(A² + B² + 2AB cosθ)

Q3. What happens if both students pull with equal force in opposite directions?

Resultant force becomes zero.

8. Statement Based Questions

Q1.

Statement I: Vectors have magnitude and direction.

Statement II: Scalars have only magnitude.

Both statements are true.

Q2.

Statement I: Resultant depends on angle between vectors.

Statement II: Resultant is independent of vector magnitudes.

Statement I is true but Statement II is false.

9. Match the Columns

Column A	Column B
1. θ = 0°	a. |A - B|
2. θ = 180°	b. A + B
3. θ = 90°	c. Pythagoras theorem
4. Resultant formula	d. Law of cosines

Answers:

1 → b
2 → a
3 → c
4 → d

10. Important Formula Sheet

R = √(A² + B² + 2AB cosθ)

tanα = (B sinθ)/(A + B cosθ)

11. Important Points for CBSE and NEET

• Resultant is maximum at 0°.
• Resultant is minimum at 180°.
• Perpendicular vectors use Pythagoras theorem.
• Law of cosines gives magnitude.
• Tangent formula gives direction.

Internal Links

Motion in a Plane Class 11 Notes

Scalars and Vectors Explained

Laws of Motion Class 11 Physics

Projectile Motion Notes

Work Energy and Power Notes

Units and Dimensions Class 11

Vector Algebra Formulas

NEET Physics Important Questions

CBSE Assertion Reason Questions Physics

Class 11 Physics Formula Sheet