Class 11 Physics Vectors Notes, MCQs and Questions with Answers

 

Subtraction of Vectors and Parallelogram Law of Vector Addition (NEET Level Notes)

- Dr.Sanjaykumar pawar

1. Subtraction of Vectors

Vector subtraction is defined using vector addition.

Suppose we have two vectors:

• A
• B

Then subtraction is written as:

\vec{A}-\vec{B}

This means:

• Add vector A with the opposite of vector B.

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2. Formula for Vector Subtraction

\vec{A}-\vec{B}=\vec{A}+(-\vec{B})

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3. Meaning of Negative Vector (−B)

The vector −B has:

• Same magnitude as B
• Opposite direction to B

Example:

• If B acts towards east,
• Then −B acts towards west.

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4. Graphical Method of Vector Subtraction

To find:

\vec{R}=\vec{A}-\vec{B}

follow these steps.

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Step 1

Draw vector A.

Step 2

Draw vector −B.

• Its length is same as B
• Direction is opposite to B

Step 3

Place tail of −B at the head of A.

Step 4

Join the tail of A to the head of −B.

The obtained vector is the resultant vector:

\vec{R}=\vec{A}-\vec{B}

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5. Important Observation

• In subtraction, we never directly subtract vectors.
• We always:
  • Reverse the direction of second vector
  • Then perform vector addition

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6. Comparison Between A + B and A − B

For Addition

\vec{R_1}=\vec{A}+\vec{B}

• B is added in same direction.

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For Subtraction

\vec{R_2}=\vec{A}-\vec{B}

• Opposite vector −B is added.

Hence resultant direction changes.

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7. Parallelogram Law of Vector Addition

Another graphical method to add vectors is:

Parallelogram Method

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8. Definition of Parallelogram Law

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

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9. Steps of Parallelogram Method

Suppose vectors are:

• A
• B

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Step 1

Draw vectors A and B from the same origin O.

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Step 2

From head of A:

• Draw a line parallel to B.

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Step 3

From head of B:

• Draw a line parallel to A.

This forms a parallelogram.

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Step 4

Draw diagonal from origin O to opposite corner S.

This diagonal OS gives the resultant vector R.

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10. Resultant Vector

The diagonal represents:

\vec{R}=\vec{A}+\vec{B}

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11. Important Point

The resultant vector:

• Starts from common origin
• Lies along diagonal of parallelogram

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12. Triangle Method vs Parallelogram Method

Both methods give the same resultant vector.

Triangle Method

• Vectors arranged head-to-tail

Parallelogram Method

• Vectors start from same origin

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13. Conclusion

Both methods are equivalent.

That means:

• Triangle law and parallelogram law produce same resultant vector.

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14. Key NEET Concepts

Vector Subtraction

\vec{A}-\vec{B}=\vec{A}+(-\vec{B})

Negative Vector

• Same magnitude
• Opposite direction

Parallelogram Law

• Resultant is diagonal of parallelogram

Triangle Law

• Resultant is third side of triangle

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15. Quick Revision Table

Concept	Formula/Meaning
Vector subtraction
Negative vector	Same magnitude, opposite direction
Resultant vector
Triangle method	Head-to-tail method
Parallelogram method	Resultant is diagonal

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16. NEET Practice Questions

Question 1

What is done first in vector subtraction?

Answer

The direction of second vector is reversed.

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Question 2

In parallelogram law, the resultant is represented by:

Answer

Diagonal of parallelogram.

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Question 3

Which methods of vector addition are equivalent?

Answer

• Triangle law
• Parallelogram law  

Internal Links

Motion in a Straight Line Class 11 Notes

Scalars and Vectors Difference

Laws of Motion Notes for NEET

Projectile Motion Complete Notes

Units and Measurements MCQs

Kinematics Formula Sheet

Work Energy and Power Notes

Physics Chapter Wise MCQs for Class 11

NEET Physics Important Questions

Class 11 Physics Revision Notes

Addition and subtraction of vectors using triangle law and parallelogram law for Class 11 Physics students.

ADDITION AND SUBTRACTION OF VECTORS

│

├── 1. Vector

│   ├── Has magnitude

│   ├── Has direction

│   └── Represented by arrow

│

├── 2. Addition of Vectors

│   │

│   ├── Resultant Vector

│   │   └── R = A + B

│   │

│   ├── Triangle Law

│   │   ├── Vectors arranged head-to-tail

│   │   ├── Third side gives resultant

│   │   └── Also called head-to-tail method

│   │

│   ├── Steps

│   │   ├── Draw vector A

│   │   ├── Place tail of B at head of A

│   │   ├── Join tail of A to head of B

│   │   └── Obtain resultant R

│   │

│   ├── Commutative Law

│   │   └── A + B = B + A

│   │

│   └── Associative Law

│       └── (A + B) + C = A + (B + C)

│

├── 3. Subtraction of Vectors

│   │

│   ├── Formula

│   │   └── A − B = A + (−B)

│   │

│   ├── Negative Vector (−B)

│   │   ├── Same magnitude as B

│   │   └── Opposite direction

│   │

│   ├── Steps

│   │   ├── Draw vector A

│   │   ├── Reverse direction of B

│   │   ├── Add −B to A

│   │   └── Get resultant R

│   │

│   └── Important Point

│       └── Subtraction is actually addition of opposite vector

│

├── 4. Parallelogram Law

│   │

│   ├── Both vectors start from same origin

│   │

│   ├── Steps

│   │   ├── Draw A and B from same point

│   │   ├── Complete parallelogram

│   │   ├── Draw diagonal

│   │   └── Diagonal gives resultant

│   │

│   └── Resultant

│       └── R = A + B

│

├── 5. Zero (Null) Vector

│   │

│   ├── Formula

│   │   └── A + (−A) = 0

│   │

│   ├── Properties

│   │   ├── Magnitude = 0

│   │   ├── No fixed direction

│   │   ├── A + 0 = A

│   │   ├── λ0 = 0

│   │   └── 0A = 0

│   │

│   └── Physical Meaning

│       └── Object returns to starting point

│

└── 6. Important NEET Points

    ├── Vector addition follows triangle law

    ├── Parallelogram and triangle laws are equivalent

    ├── Vector addition is commutative

    ├── Vector addition is associative

    ├── Opposite vectors give zero vector

    └── Subtraction uses opposite vector

CBSE Class 11 Physics

Addition and Subtraction of Vectors

Question Bank with Answers

1. Multiple Choice Questions (MCQs)

Q1. A vector has:

• a) Only magnitude
• b) Only direction
• c) Magnitude and direction
• d) Neither magnitude nor direction

Answer: c) Magnitude and direction

Q2. The graphical method of vector addition is called:

• a) Rectangle law
• b) Head-to-tail method
• c) Dot method
• d) Circular method

Answer: b) Head-to-tail method

Q3. The resultant of equal and opposite vectors is:

• a) Unit vector
• b) Scalar
• c) Zero vector
• d) Parallel vector

Answer: c) Zero vector

2. Very Short Answer Questions

Q1. Define vector.

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

Q2. What is a zero vector?

A vector having zero magnitude is called a zero vector.

Q3. Write formula for vector subtraction.

A − B = A + (−B)

3. Short Answer Questions

Q1. Explain triangle law of vector addition.

According to triangle law, if two vectors are represented by two sides of a triangle taken in order, then the third side taken in opposite order gives the resultant vector.

Q2. What is head-to-tail method?

In this method, the tail of second vector is placed at the head of first vector to obtain resultant vector.

4. Long Answer Questions

Q1. Explain parallelogram law of vector addition.

According to parallelogram law, if two vectors acting simultaneously are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal passing through common origin.

Steps:

1. Draw vectors A and B from same origin.
2. Complete the parallelogram.
3. Draw diagonal from common origin.
4. The diagonal gives resultant vector.

R = A + B

5. Assertion and Reason Questions

Assertion (A): Vector addition is commutative.
Reason (R): A + B = B + A

Both Assertion and Reason are true and Reason correctly explains Assertion.

Assertion (A): Zero vector has definite direction.
Reason (R): Zero vector has zero magnitude.

Assertion is false but Reason is true.

6. Fill in the Blanks

1. A vector has magnitude and ________.

direction

2. Vector subtraction is defined using vector ________.

addition

3. Resultant in parallelogram law is represented by ________.

diagonal

7. Case Study Questions

Ravi walks 4 m east and then 3 m north. The displacement is represented using vectors.

Q1. Which law is used here?

Triangle law of vector addition.

Q2. What is obtained after addition?

Resultant vector.

Q3. Which graphical method is used?

Head-to-tail method.

8. Statement Based Questions

Statement	Answer
Vector addition depends on order.	False
Opposite vectors have same magnitude.	True
Zero vector has infinite magnitude.	False
Diagonal of parallelogram gives resultant.	True

9. Match the Columns

Column A	Column B
Zero vector	Zero magnitude
Triangle law	Head-to-tail method
Negative vector	Opposite direction
Parallelogram law	Diagonal gives resultant

10. Important Formulae

R = A + B

A − B = A + (−B)

A + B = B + A

(A + B) + C = A + (B + C)

A + (−A) = 0