Resolution of Vectors Class 11 Physics Notes, MCQs & Examples (NEET + CBSE)

 RESOLUTION OF VECTORS

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├── 1. Meaning

│   ├── Splitting a vector into components

│   ├── Components add to form original vector

│   └── Used in force, motion, displacement problems

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├── 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

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├── 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.

 

- Dr.Sanjaykumar pawar

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.

Example: A diagonal force can be divided into horizontal and vertical components.

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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:

A = λa + μb

Where:

• λ and μ are real numbers
• λa and μb are component vectors

Vector A is said to be resolved into components along vectors a and b.

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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

|î| = |ĵ| = |k̂| = 1

Unit vectors have no dimensions and no units.

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4. Vector in Terms of Unit Vector

Any vector can be written as:

A = |A| n̂

Where:

• |A| = magnitude of vector
• n̂ = unit vector in direction of A

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5. Resolution Along x and y Axes

A vector A in a plane can be resolved into x-component and y-component.

A = Ax î + Ay ĵ

Where:

• Ax = x-component
• Ay = y-component

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6. Formula for Components

If vector A makes angle θ with x-axis:

Ax = A cosθ

Ay = A sinθ

x-component uses cosine and y-component uses sine.

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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

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8. Magnitude of Vector

If Ax and Ay are known:

A = √(Ax² + Ay²)

This formula is based on Pythagoras theorem.

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9. Direction of Vector

tanθ = Ay / Ax

Therefore:

θ = tan⁻¹(Ay / Ax)

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10. Two Ways to Represent a Vector

Method 1

• Magnitude A
• Direction θ

Method 2

• x-component Ax
• y-component Ay

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11. Resolution in Three Dimensions

In 3D, a vector has three components:

Ax = A cosα

Ay = A cosβ

Az = A cosγ

Where:

• α = angle with x-axis
• β = angle with y-axis
• γ = angle with z-axis

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12. Vector Form in 3D

A = Ax î + Ay ĵ + Az k̂

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13. Magnitude in 3D

A = √(Ax² + Ay² + Az²)

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14. Position Vector

A position vector is written as:

r = x î + y ĵ + z k̂

Where x, y, z are coordinates of the point.

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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

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16. Common NEET Mistakes

• Forgetting signs of components
• Confusing sine and cosine
• Ignoring quadrant rules
• Writing scalar as vector

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17. Quick Trick for NEET

• Cos → adjacent side
• Sin → opposite side
• x-component → cosine
• y-component → sine

Prepared for NEET Physics Students

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