RESOLUTION OF VECTORS
│
├── 1. Meaning
│ ├── Splitting a vector into components
│ ├── Components add to form original vector
│ └── Used in force, motion, displacement problems
│
├── 2. Vector Resolution in Plane
│ ├── Vector A resolved along vectors a and b
│ ├── Formula:
│ │ A = λa + μb
│ ├── λ and μ are real numbers
│ └── Component vectors:
│ ├── λa
│ └── μb
│
├── 3. Unit Vectors
│ ├── Magnitude = 1
│ ├── Show direction only
│ ├── No dimension or unit
│ ├── Along axes:
│ │ ├── î → x-axis
│ │ ├── ĵ → y-axis
│ │ └── k̂ → z-axis
│ └── Properties:
│ ├── |î| = |ĵ| = |k̂| = 1
│ └── Mutually perpendicular
│
├── 4. Vector in Unit Vector Form
│ ├── Formula:
│ │ A = |A| n̂
│ ├── |A| → magnitude
│ └── n̂ → unit vector along A
│
├── 5. Resolution Along x and y Axes
│ ├── Vector A in x-y plane
│ ├── Components:
│ │ ├── Ax along x-axis
│ │ └── Ay along y-axis
│ └── Vector form:
│ A = Ax î + Ay ĵ
│
├── 6. Component Formulae
│ ├── Ax = A cosθ
│ ├── Ay = A sinθ
│ └── θ = angle with x-axis
│
├── 7. Nature of Components
│ ├── Positive
│ ├── Negative
│ └── Zero
│
├── 8. Magnitude of Vector
│ └── Formula:
│ A = √(Ax² + Ay²)
│
├── 9. Direction of Vector
│ ├── tanθ = Ay/Ax
│ └── θ = tan⁻¹(Ay/Ax)
│
├── 10. Ways to Represent Vector
│ ├── By magnitude and direction
│ │ ├── A
│ │ └── θ
│ └── By components
│ ├── Ax
│ └── Ay
│
├── 11. Resolution in 3D
│ ├── Components:
│ │ ├── Ax = A cosα
│ │ ├── Ay = A cosβ
│ │ └── Az = A cosγ
│ ├── Vector form:
│ │ A = Ax î + Ay ĵ + Az k̂
│ └── Magnitude:
│ A = √(Ax² + Ay² + Az²)
│
├── 12. Position Vector
│ └── r = xî + yĵ + zk̂
│
├── 13. NEET Important Points
│ ├── x-component → cosine
│ ├── y-component → sine
│ ├── Components are scalars
│ └── Axî and Ayĵ are vectors
│
└── 14. Common Mistakes
├── Wrong sign of components
├── Confusing sin and cos
├── Ignoring quadrant
└── Treating Ax as vector
 |
| Resolution of a vector into x and y components showing Ax = A cosθ and Ay = A sinθ on Cartesian axes. |
Internal Links
/class-11-physics-vectors
/motion-in-a-plane-notes
/neet-physics-important-formulas
/physics-mcqs-class-11
/unit-vectors-and-components
/cbse-class-11-physics-notes
/physics-numericals-practice-set
Resolution of Vectors – NEET Notes
1. Meaning of Resolution of Vectors
Resolution of a vector means splitting a vector into two or more parts called components.
These components combine together to form the original vector.
2. Resolving a Vector in a Plane
Let there be two vectors a and b in the same plane. Another vector A can be written as:
Where:
- λ and μ are real numbers
- λa and μb are component vectors
3. Unit Vectors
A unit vector is a vector having magnitude equal to 1.
It is used only to represent direction.
Unit Vectors Along Coordinate Axes
- Along x-axis → î
- Along y-axis → ĵ
- Along z-axis → k̂
4. Vector in Terms of Unit Vector
Any vector can be written as:
Where:
- |A| = magnitude of vector
- n̂ = unit vector in direction of A
5. Resolution Along x and y Axes
A vector A in a plane can be resolved into x-component and y-component.
Where:
- Ax = x-component
- Ay = y-component
6. Formula for Components
If vector A makes angle θ with x-axis:
7. Sign of Components
Components can be positive, negative, or zero depending on direction.
| Quadrant | x-component | y-component |
|---|---|---|
| First | Positive | Positive |
| Second | Negative | Positive |
| Third | Negative | Negative |
| Fourth | Positive | Negative |
8. Magnitude of Vector
If Ax and Ay are known:
This formula is based on Pythagoras theorem.
9. Direction of Vector
Therefore:
10. Two Ways to Represent a Vector
Method 1
- Magnitude A
- Direction θ
Method 2
- x-component Ax
- y-component Ay
11. Resolution in Three Dimensions
In 3D, a vector has three components:
Where:
- α = angle with x-axis
- β = angle with y-axis
- γ = angle with z-axis
12. Vector Form in 3D
13. Magnitude in 3D
14. Position Vector
A position vector is written as:
Where x, y, z are coordinates of the point.
15. NEET Important Points
- Resolution means splitting vectors into components.
- Unit vectors show direction only.
- î, ĵ, k̂ are unit vectors.
- Ax = A cosθ
- Ay = A sinθ
- A = √(Ax² + Ay²)
- tanθ = Ay / Ax
16. Common NEET Mistakes
- Forgetting signs of components
- Confusing sine and cosine
- Ignoring quadrant rules
- Writing scalar as vector
17. Quick Trick for NEET
- Cos → adjacent side
- Sin → opposite side
- x-component → cosine
- y-component → sine
Class 11 Physics - Resolution of Vectors
1. Multiple Choice Questions (MCQs)
Q1. Resolution of a vector means:
(a) Adding vectors
(b) Splitting into components
(c) Multiplying vectors
(d) Rotating vectors
Answer: (b) Splitting into components
Q2. Unit vector has magnitude:
(a) 0 (b) 1 (c) 2 (d) infinite
Answer: (b) 1
Q3. x-component of vector A is:
(a) A sinθ (b) A cosθ (c) A tanθ (d) A cotθ
Answer: (b) A cosθ
Q4. Unit vector along x-axis is:
Answer: î
2. Very Short Answer Questions
Q1. Define resolution of vectors.
Answer: Splitting a vector into components along different directions.
Q2. What is unit vector?
Answer: A vector with magnitude 1 that shows direction only.
Q3. Name unit vectors.
Answer: î, ĵ, k̂
3. Short Answer Questions
Q1. Write vector in component form.
Answer: A = Ax î + Ay ĵ
Q2. Why are components useful?
Answer: They simplify calculations in physics problems.
4. Long Answer Questions
Q1. Explain resolution of vector in 2D.
A vector A making angle θ with x-axis can be resolved into components:
Ax = A cosθ
Ay = A sinθ
A = Ax î + Ay ĵ
Magnitude: A = √(Ax² + Ay²)
5. Assertion and Reason
Q1.
Assertion: Unit vectors have magnitude 1.
Reason: They are used to represent direction only.
Answer: Both are true and Reason is correct explanation.
6. Fill in the Blanks
Q1. Resolution means splitting vector into ______.
Answer: components
Q2. Unit vector along y-axis is ______.
Answer: ĵ
7. Match the Column
î → x-axis
ĵ → y-axis
k̂ → z-axis
Ax → x-component
8. Case Study
A force of 20 N makes 30° with x-axis.
Q1. Find Ax
Answer: Ax = 20 cos30 = 10√3 N
Q2. Find Ay
Answer: Ay = 20 sin30 = 10 N
9. Numericals
Q1. Find magnitude of vector (3,4)
Answer: √(3² + 4²) = 5
