Equality of Vectors Class 11 Notes for NEET & CBSE Students

 

Equality of Vectors — Easy Notes for NEET  

- Dr.Sanjaykumar pawar

Definition of Equal Vectors

Two vectors A and B are said to be equal vectors when:

1. They have the same magnitude (length)
2. They have the same direction

👉 Both conditions must be satisfied.

---

Important Point

Even if two vectors are placed at different positions, they can still be equal if their:

• lengths are equal
• directions are the same

The position of a vector does not matter.

---

Line-by-Line Explanation

“Two vectors A and B are said to be equal if, and only if, they have the same magnitude and the same direction.”

Easy Meaning:

Vectors are equal only when:

• their lengths are equal
• and they point in the same direction

Example:

If vector A is 5 units toward east and vector B is also 5 units toward east, then:

A = B

---

“Figure 3.2(a) shows two equal vectors A and B.”

Easy Meaning:

The figure shows two vectors that are equal because:

• both have equal length
• both point in the same direction

---

“We can easily check their equality.”

Easy Meaning:

We can test whether two vectors are equal or not.

---

“Shift B parallel to itself until its tail Q coincides with that of A.”

Easy Meaning:

Move vector B without changing:

• its direction
• its size

Bring the starting point (tail) of B to the starting point of A.

👉 This process is called parallel shifting.

---

“Then, since their tips S and P also coincide, the two vectors are said to be equal.”

Easy Meaning:

After shifting:

• if the ending points (tips) also meet,
• then both vectors are equal.

✅ Same start point + same end point = equal vectors.

---

“In general, equality is indicated as A = B.”

Easy Meaning:

Equal vectors are written as:

\vec{A} = \vec{B}

---

“Note that in Fig. 3.2(b), vectors A′ and B′ have the same magnitude but they are not equal because they have different directions.”

Easy Meaning:

In this figure:

• both vectors have equal length
• but they point in different directions

So they are not equal vectors.

❌ Same length alone is not enough.

---

“Even if we shift B′ parallel to itself so that its tail Q′ coincides with the tail O′ of A′, the tip S′ of B′ does not coincide with the tip P′ of A′.”

Easy Meaning:

After moving vector B′:

• the starting points become same
• but the ending points do not match

This happens because directions are different.

Therefore:

\vec{A'} \ne \vec{B'}

---

Quick Revision for NEET

Conditions for Equal Vectors

✅ Same magnitude
✅ Same direction

---

Not Necessary

❌ Same position

---

Shortcut Trick

Think:

“Length + Direction both same = Equal vectors”

---

NEET Important Points

• Vectors can be shifted parallel to themselves.
• Shifting does not change a vector.
• Equal vectors may start from different points.
• Vectors with same magnitude but different directions are unequal.

---

Example Questions

Example 1

Two vectors each of magnitude 4 N acting toward north are:

✅ Equal vectors

---

Example 2

Two vectors each of magnitude 5 m but one toward east and one toward west are:

❌ Not equal vectors

---

One-Line Summary

👉 Equal vectors have the same magnitude and same direction, regardless of their position.   

Equality of vectors showing same magnitude and same direction in Class 11 Physics.

CBSE Class 11 Physics — Equality of Vectors Question Bank

Multiple Choice Questions (MCQs)

1. Two vectors are equal when they have:

a) Same magnitude only
b) Same direction only
c) Same magnitude and same direction
d) Same initial point

✅ Answer: c) Same magnitude and same direction

---

2. Equal vectors may have:

a) Different magnitudes
b) Different directions
c) Different positions
d) Different lengths

✅ Answer: c) Different positions

---

3. If two vectors have same magnitude but opposite directions, they are:

a) Equal vectors
b) Unequal vectors
c) Unit vectors
d) Zero vectors

✅ Answer: b) Unequal vectors

---

4. Shifting a vector parallel to itself:

a) Changes magnitude
b) Changes direction
c) Does not change the vector
d) Makes it zero

✅ Answer: c) Does not change the vector

---

5. Which of the following is necessary for equality of vectors?

a) Same tail position
b) Same head position
c) Same magnitude and direction
d) Same line only

✅ Answer: c) Same magnitude and direction

---

Very Short Answer Questions (1 Mark)

1. Define equal vectors.

✅ Answer:
Two vectors having the same magnitude and same direction are called equal vectors.

---

2. Can equal vectors have different initial points?

✅ Answer:
Yes, equal vectors may have different initial points.

---

3. What happens when a vector is shifted parallel to itself?

✅ Answer:
Its magnitude and direction remain unchanged.

---

4. Are vectors with same magnitude always equal?

✅ Answer:
No, they must also have the same direction.

---

5. Write the symbol used for equality of vectors.

✅ Answer:

\vec{A} = \vec{B}

---

Short Answer Questions (2–3 Marks)

1. State the conditions for two vectors to be equal.

✅ Answer: Two vectors are equal if:

1. Their magnitudes are equal.
2. Their directions are the same.

Their positions may be different.

---

2. Explain why two vectors of same length may not be equal.

✅ Answer: Two vectors with same length may not be equal because their directions may differ. Equal vectors must have both same magnitude and same direction.

---

3. What is meant by shifting a vector parallel to itself?

✅ Answer: Moving a vector without changing its magnitude and direction is called shifting parallel to itself. The vector remains unchanged after shifting.

---

Long Answer Questions (5 Marks)

1. Explain the equality of vectors with the help of a diagram.

✅ Answer: Two vectors are said to be equal when they have:

• same magnitude
• same direction

To check equality, one vector is shifted parallel to itself so that its tail coincides with the tail of the other vector.

If the tips also coincide, the vectors are equal.

Equal vectors are represented as:

\vec{A} = \vec{B}

If vectors have same magnitude but different directions, they are unequal.

---

2. Differentiate between equal and unequal vectors.

✅ Answer:

Equal Vectors	Unequal Vectors
Same magnitude	Magnitude may or may not be same
Same direction	Different direction
Tips coincide after shifting	Tips do not coincide
Represented by A = B	Represented by A ≠ B

---

Assertion and Reason Questions

1.

Assertion (A): Two vectors with same magnitude are always equal.
Reason (R): Equal vectors must have same direction also.

✅ Answer: Assertion is false, Reason is true.

---

2.

Assertion (A): Equal vectors can have different positions.
Reason (R): Position does not affect equality of vectors.

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

---

3.

Assertion (A): Parallel shifting changes the vector.
Reason (R): Magnitude changes during shifting.

✅ Answer: Both Assertion and Reason are false.

---

Fill in the Blanks

1. Two vectors are equal if they have same ______ and same direction.
   ✅ Answer: magnitude

---

2. Equal vectors may have different ______.
   ✅ Answer: positions

---

3. A vector shifted parallel to itself remains ______.
   ✅ Answer: unchanged

---

4. Vectors with same magnitude but different directions are ______ vectors.
   ✅ Answer: unequal

---

5. Equality of vectors is written as ______.
   ✅ Answer:

\vec{A} = \vec{B}

---

Statement-Based Questions

1. Statement I:

Equal vectors must have equal lengths.

Statement II: Equal vectors must point in the same direction.

a) Both statements are true
b) Both statements are false
c) Statement I true, II false
d) Statement I false, II true

✅ Answer: a) Both statements are true

---

2. Statement I:

Position determines equality of vectors.

Statement II: Direction is important for vector equality.

✅ Answer: Statement I is false, Statement II is true.

---

Match the Columns

Column A	Column B
1. Equal vectors	a. Different direction
2. Unequal vectors	b. Same magnitude and direction
3. Parallel shifting	c. No change in vector
4. Same magnitude only	d. Not equal

✅ Answers: 1 → b
2 → a
3 → c
4 → d

---

Case Study Questions

Case Study

Rahul draws two vectors A and B of length 5 cm pointing east. He places them at different positions on paper. After shifting vector B parallel to itself, both vectors completely overlap.

Questions

1. Are vectors A and B equal?

✅ Answer:
Yes, because they have same magnitude and same direction.

---

2. Does position affect equality of vectors?

✅ Answer:
No, position does not affect equality.

---

3. What happens after parallel shifting?

✅ Answer:
The vector remains unchanged.

---

4. If vector B pointed west instead of east, would the vectors be equal?

✅ Answer:
No, because directions would be different. 

Internal Links

Introduction to Vectors Class 11

Types of Vectors Notes

Vector Addition and Subtraction

Triangle Law of Vector Addition

Scalar and Vector Quantities

Resolution of Vectors

Unit Vector Notes

Motion in a Plane Class 11

NCERT Solutions for Vector Algebra

NEET Physics Important Questions

---

Important CBSE One-Liners

• Equal vectors have same magnitude and direction.
• Position does not affect vector equality.
• Parallel shifting does not change a vector.
• Same magnitude alone is not sufficient for equality.

Equality of Vectors

│

├── Definition

│   ├── Same magnitude

│   └── Same direction

│

├── Equal Vectors

│   ├── Length equal

│   ├── Direction same

│   ├── Position may be different

│   └── Written as A = B

│

├── Checking Equality

│   ├── Shift vector parallel to itself

│   ├── Tails coincide

│   └── Tips also coincide

│       └── Vectors are equal

│

├── Unequal Vectors

│   ├── Same magnitude

│   ├── Different directions

│   └── Therefore not equal

│

├── Important Points

│   ├── Shifting does not change vector

│   ├── Direction is very important

│   └── Same length alone is not enough

│

├── Example of Equal Vectors

│   ├── 5 N east

│   └── 5 N east

│

├── Example of Unequal Vectors

│   ├── 5 m east

│   └── 5 m west

│

└── NEET Shortcut

    └── Same Length + Same Direction = Equal Vectors