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.
Step 2: Resolve vector B into components
Draw perpendicular SN on OP.
Now vector B has two components:
Horizontal Component
Vertical Component
Step 3: Find ON and SN
From geometry:
But,
Therefore,
Also,
Step 4: Apply Pythagoras Theorem
In right triangle OSN:
Substituting values:
Step 5: Expand the Equation
Take B² common:
Using identity:
Therefore,
Final Formula for Magnitude
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:
Also,
Equating both:
Therefore,
Step 7: Formula for tan α
From triangle:
Substitute values:
Important Results for NEET
Magnitude of Resultant
Direction of Resultant
Special Cases
1. When θ = 0°
2. When θ = 180°
3. When θ = 90°
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)
Q2. If two vectors act in the same direction, the resultant is:
A) A - B
B) A + B
C) AB
D) Zero
Q3. If angle between two vectors is 180°, the resultant is:
A) A + B
B) |A - B|
C) Zero
D) AB
2. Very Short Answer Questions
Q1. What is a resultant vector?
Q2. Which method is used to add two vectors geometrically?
Q3. What is the resultant when two vectors are perpendicular?
Q4. At which angle is resultant maximum?
3. Short Answer Questions
Q1. State parallelogram law of vector addition.
Q2. Write formula for magnitude of resultant vector.
- A and B are magnitudes of vectors
- θ is angle between them
Q3. Write formula for direction of resultant vector.
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:
Apply Pythagoras theorem:
Expanding:
Using identity:
Therefore:
Hence,
Q2. Derive formula for direction of resultant vector.
From the triangle:
Where:
Therefore:
5. Assertion and Reason Questions
Q1.
Assertion (A): Resultant of two equal and opposite vectors is zero.
Reason (R): Opposite vectors cancel each other.
Q2.
Assertion (A): Resultant of perpendicular vectors is equal to sum of vectors.
Reason (R): Pythagoras theorem is used for perpendicular vectors.
6. Fill in the Blanks
Q1. The diagonal of parallelogram gives the _________ vector.
Q2. The formula for resultant vector uses law of _________.
Q3. If θ = 0°, resultant is _________.
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?
Q2. Write formula for resultant force.
Q3. What happens if both students pull with equal force in opposite directions?
8. Statement Based Questions
Q1.
Statement I: Vectors have magnitude and direction.
Statement II: Scalars have only magnitude.
Q2.
Statement I: Resultant depends on angle between vectors.
Statement II: Resultant is independent of vector magnitudes.
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 |
1 → b
2 → a
3 → c
4 → d
10. Important Formula Sheet
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.
