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