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Resolution of Vectors
The process of splitting a vector into two or more component vectors along specified directions is called the resolution of a vector.
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Resolution of Vectors
Learning Outcomes
- Define the resolution of a vector.
- Explain rectangular components of a vector.
- Resolve a vector along the horizontal and vertical axes.
- Select sine or cosine according to the given angle.
- Determine the signs of vector components.
- Write a vector using unit-vector notation.
- Calculate a vector’s magnitude from its components.
- Determine its direction and correct quadrant.
- Apply vector resolution to forces, motion, and inclined planes.
If a vector makes an angle with the horizontal, use: Horizontal component = A cos θ Vertical component = A sin θ
Recommended for Class 11
Resolution of Vectors
Chapter 2 — Vectors · Topic: Resolution of Vectors
Learning Outcomes
- Define the resolution of a vector.
- Explain rectangular components of a vector.
- Resolve a vector along the horizontal and vertical axes.
- Select sine or cosine according to the given angle.
- Determine the signs of vector components.
- Write a vector using unit-vector notation.
- Calculate a vector’s magnitude from its components.
- Determine its direction and correct quadrant.
- Apply vector resolution to forces, motion, and inclined planes.
Definition
Introduction
A vector may act at an angle instead of acting completely horizontally or vertically. Such a vector can be difficult to use directly in calculations. Therefore, it is resolved into components along convenient coordinate axes. For a two-dimensional vector: A⃗ = A⃗ₓ + A⃗ᵧ Here: A⃗ₓ is the horizontal component. A⃗ᵧ is the vertical component. Together, these components have the same effect as the original vector.
Key Terms
Key Terms
| Term | Meaning |
|---|---|
| Vector | A physical quantity possessing both magnitude and direction. |
| Component | A part of a vector acting along a selected direction. |
| Rectangular Components | Two vector components acting perpendicular to each other. |
| Horizontal Component | The component acting parallel to the x-axis. |
| Vertical Component | The component acting parallel to the y-axis. |
| Resultant Vector | The single vector having the combined effect of its components. |
| Unit Vector | A vector of magnitude one used to represent a particular direction. |
| Reference Angle | The angle used to calculate the components of a vector. |
Detailed Concept Explanation
Suppose a vector A⃗ of magnitude A makes an angle θ with the positive horizontal axis. The vector can be resolved into: Aₓ = A cos θ and: Aᵧ = A sin θ The horizontal component is adjacent to the given angle. Therefore, cosine is used. The vertical component is opposite the given angle. Therefore, sine is used. The original vector can then be written as: A⃗ = Aₓ î + Aᵧ ĵ Therefore: A⃗ = A cos θ î + A sin θ ĵ
Concept in Easy Words
Imagine that an object is being pulled by a force acting upward at an angle. Part of the force pulls the object horizontally. The remaining part pulls it vertically. These two parts are called the vector’s components. They are not additional forces. They are simply another way of representing the original force.
Beginner, Intermediate and Advanced Explanation
Important Facts
Important Facts
- Vector components are also vectors.
- Rectangular components are perpendicular to each other.
- The components together are equivalent to the original vector.
- The horizontal component is not always A cos θ.
- The component adjacent to the given angle uses cosine.
- The component opposite the given angle uses sine.
- Component signs depend on direction and quadrant.
- A component can be negative even though the vector’s magnitude is positive.
- A vector’s magnitude is never negative.
Formulas and Mathematical Explanation
When the Angle Is Measured From the Horizontal Axis
When the Angle Is Measured From the Vertical Axis
Aₓ = A sin θ Aᵧ = A cos θ
Vector in Unit-Vector Form
Magnitude From Components
Direction From Components
tan θ = Aᵧ / Aₓ θ = tan⁻¹(Aᵧ / Aₓ) The final direction must be adjusted according to the quadrant.
Symbols and Units
Symbols and Units
| Quantity | Symbol | SI unit |
|---|---|---|
| Original vector magnitude | A | Depends on the quantity |
| Horizontal component | Aₓ | Same as A |
| Vertical component | Aᵧ | Same as A |
| Angle | θ | degree or radian |
| Horizontal unit vector | î | No unit |
| Vertical unit vector | ĵ | No unit |
| Third-axis unit vector | k̂ | No unit |
Components have the same unit as their original vector. For example, if force is measured in newtons, its components are also measured in newtons.
Signs of Components
Component signs by direction
| Direction | Component sign |
|---|---|
| Right | +x |
| Left | −x |
| Upward | +y |
| Downward | −y |
Quadrant I Aₓ > 0, Aᵧ > 0 Quadrant II Aₓ < 0, Aᵧ > 0 Quadrant III Aₓ < 0, Aᵧ < 0 Quadrant IV Aₓ > 0, Aᵧ < 0
Comparison Table
Original vector versus components
| Feature | Original vector | Horizontal component | Vertical component |
|---|---|---|---|
| Symbol | A⃗ | A⃗ₓ | A⃗ᵧ |
| Direction | Inclined | Along x-axis | Along y-axis |
| Magnitude | A | A cos θ | A sin θ |
| Unit | Depends on quantity | Same as original | Same as original |
| Nature | Vector | Vector | Vector |
Solved Examples
Solved Example 1: Resolving a Force
A force of 20 N acts at an angle of 30° above the horizontal. Find its horizontal and vertical components.
- F = 20 N
- θ = 30°
Fₓ = F cos θ
Fₓ = 20 cos 30°
Fₓ = 20(0.866)
Fₓ = 17.32 N
Fᵧ = F sin θ
Fᵧ = 20 sin 30°
Fᵧ = 20(0.5)
Fᵧ = 10 N
Key Step: The angle is measured from the horizontal, so cosine is used for the horizontal component.
Final answer: Fₓ = 17.32 N; Fᵧ = 10 N
Solved Example 2: Angle Measured From the Vertical
A velocity of 15 m s⁻¹ makes an angle of 40° with the vertical. Find its horizontal and vertical components.
- v = 15 m s⁻¹
- θ = 40°
The horizontal component is opposite the given angle.
vₓ = v sin θ
vₓ = 15 sin 40°
vₓ = 15(0.643)
vₓ = 9.65 m s⁻¹
The vertical component is adjacent to the given angle.
vᵧ = v cos θ
vᵧ = 15 cos 40°
vᵧ = 15(0.766)
vᵧ = 11.49 m s⁻¹
Final answer: vₓ = 9.65 m s⁻¹; vᵧ = 11.49 m s⁻¹
Solved Example 3: Vector in the Second Quadrant
A force of 50 N makes an angle of 120° with the positive x-axis. Determine its components.
- F = 50 N
- θ = 120°
Fₓ = F cos θ = 50 cos 120° = 50(−0.5) = −25 N
Fᵧ = F sin θ = 50 sin 120° = 50(0.866) = 43.3 N
F⃗ = Fₓ î + Fᵧ ĵ
F⃗ = −25 î + 43.3 ĵ N
Key Step: The force lies in Quadrant II, so its horizontal component is negative and its vertical component is positive.
Final answer: Fₓ = −25 N; Fᵧ = 43.3 N
Solved Example 4: Magnitude From Components
A vector has components Aₓ = 6 N and Aᵧ = 8 N. Find its magnitude.
- Aₓ = 6 N
- Aᵧ = 8 N
A = √(Aₓ² + Aᵧ²)
A = √((6)² + (8)²)
A = √(36 + 64)
A = √100
A = 10 N
Final answer: A = 10 N
Solved Example 5: Magnitude and Direction
A vector has components Aₓ = −6 N and Aᵧ = 8 N. Find its magnitude and direction measured anticlockwise from the positive x-axis.
- Aₓ = −6 N
- Aᵧ = 8 N
A = √(Aₓ² + Aᵧ²) = √((−6)² + (8)²) = √(36 + 64) = 10 N
α = tan⁻¹(|Aᵧ| / |Aₓ|) = tan⁻¹(86) = 53.13°
The vector lies in Quadrant II.
θ = 180° − α = 180° − 53.13° = 126.87°
Final answer: A = 10 N; θ = 126.87°
Resolution of Weight on an Inclined Plane
For an object of mass m placed on a plane inclined at angle θ: Component Parallel to the Plane W∥ = mg sin θ This component pulls the object down the slope. Component Perpendicular to the Plane W⊥ = mg cos θ This component presses the object against the surface. If there is no acceleration perpendicular to the plane: N = mg cos θ Here, N is the normal reaction.
Solved Example 6: Weight on an Inclined Plane
A 5 kg object is placed on a smooth plane inclined at 30°. Calculate the components of its weight parallel and perpendicular to the plane. Take g = 9.8 m s⁻².
- m = 5 kg
- θ = 30°
- g = 9.8 m s⁻²
W = mg = 5(9.8) = 49 N
W∥ = W sin θ = 49 sin 30° = 49(0.5) = 24.5 N
W⊥ = W cos θ = 49 cos 30° = 49(0.866) = 42.43 N
Final answer: W∥ = 24.5 N; W⊥ = 42.43 N
Real-Life Applications
Vector resolution is used in
- Projectile motion
- Pulling an object with an inclined rope
- Pushing a lawn roller
- Motion on inclined planes
- Aircraft navigation
- Boat movement in flowing water
- Force analysis in bridges
- Tension in cables
- Static equilibrium
- Circular motion
- Wind velocity calculations
- Electric and magnetic forces
Key Logic
Tips and Tricks
Tips and Tricks
- Identify the reference axis before selecting sine or cosine.
- Use cosine for the component adjacent to the given angle.
- Use sine for the component opposite the given angle.
- Determine the component signs from their directions.
- Keep the calculator in degree mode when angles are given in degrees.
- Draw axes in rough working when necessary, but no diagram is required in the final notes.
- Check the result using: A = √(Aₓ² + Aᵧ²)
- Each component’s magnitude should not exceed the original vector’s magnitude.
- Always write the same unit as the original vector.
Common Mistakes
Common Mistakes
- Using sine for the adjacent component.
- Using cosine for the opposite component.
- Assuming the angle is always measured from the horizontal.
- Ignoring negative component signs.
- Giving a negative value for vector magnitude.
- Using tan⁻¹(Aᵧ/Aₓ) without checking the quadrant.
- Mixing force and velocity units.
- Adding component magnitudes directly.
- Forgetting the square root when calculating magnitude.
- Treating components as unrelated vectors.
Topic Summary
Vector resolution is the process of expressing a vector through components along selected directions. In two dimensions, rectangular components normally act along the x-axis and y-axis. Sine and cosine are selected according to whether a component is opposite or adjacent to the given angle. Correct signs and quadrant identification are essential in numerical problems.
Exam Facts
Exam Facts
- Resolution is the reverse process of vector addition.
- Rectangular components act perpendicular to each other.
- When the angle is measured from the horizontal: Aₓ = A cos θ and Aᵧ = A sin θ
- Components have the same physical unit as the original vector.
- A negative component indicates direction, not negative magnitude.
- The magnitude is calculated using the Pythagorean theorem.
- Direction is calculated using inverse tangent and quadrant information.
- On an inclined plane: W∥ = mg sin θ and W⊥ = mg cos θ
Three-Dimensional Components
In three dimensions, a vector may have components along all three coordinate axes: A⃗ = Aₓ î + Aᵧ ĵ + Az k̂ Its magnitude is: A = √(Aₓ² + Aᵧ² + Az²)
Exam Focus
Exam Focus
Important points
- Resolution is the reverse process of vector addition.
- Rectangular components act perpendicular to each other.
- When the angle is measured from the horizontal: Aₓ = A cos θ and Aᵧ = A sin θ
- Components have the same physical unit as the original vector.
- A negative component indicates direction, not negative magnitude.
- The magnitude is calculated using the Pythagorean theorem.
- Direction is calculated using inverse tangent and quadrant information.
- On an inclined plane: W∥ = mg sin θ and W⊥ = mg cos θ
Important formulas
- Aₓ = A cos θ (θ from horizontal)
- Aᵧ = A sin θ (θ from horizontal)
- Aₓ = A sin θ (θ from vertical)
- Aᵧ = A cos θ (θ from vertical)
- A⃗ = Aₓ î + Aᵧ ĵ
- A = √(Aₓ² + Aᵧ²)
- θ = tan⁻¹(Aᵧ / Aₓ)
- W∥ = mg sin θ
- W⊥ = mg cos θ
Common exam traps
- Using sine for the adjacent component.
- Using cosine for the opposite component.
- Assuming the angle is always measured from the horizontal.
- Ignoring negative component signs.
- Giving a negative value for vector magnitude.
- Using tan⁻¹(Aᵧ/Aₓ) without checking the quadrant.
- Adding component magnitudes directly.
- Forgetting the square root when calculating magnitude.
Short question tips
- What are rectangular components?
- Can a vector component be negative?
- Why are vectors resolved into components?
Long question structure
Consider a vector A⃗ acting at angle θ with the positive horizontal axis. The vector can be divided into two perpendicular components. The horizontal component acts along the x-axis. It is represented by Aₓ. The vertical component acts along the y-axis. It is represented by Aᵧ. The horizontal component is adjacent to angle θ. Therefore: Aₓ = A cos θ The vertical component is opposite to angle θ. Therefore: Aᵧ = A sin θ The vector can be expressed using unit vectors. Its unit-vector form is: A⃗ = Aₓ î + Aᵧ ĵ Substituting the component formulas: A⃗ = A cos θ î + A sin θ ĵ The magnitude can be recovered from the components. It is calculated using: A = √(Aₓ² + Aᵧ²) Its reference angle can be calculated using: θ = tan⁻¹(Aᵧ / Aₓ) The signs of the components depend on the vector’s quadrant. Therefore, direction and quadrant must be checked before giving the final answer.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Split the vector into perpendicular components.
Identify the axis from which the angle is measured.
Adjacent means cosine.
Opposite means sine.
The process of splitting a vector into two or more component vectors along specified directions is called the resolution of a vector.
From horizontal: Aₓ=A cos θ, Aᵧ=A sin θ; From vertical: Aₓ=A sin θ, Aᵧ=A cos θ; A=√(Aₓ²+Aᵧ²); θ=tan⁻¹(Aᵧ/Aₓ)
- Definition
- The process of splitting a vector into two or more component vectors along specified directions is called the resolution of a vector.
- Formula
- From horizontal: Aₓ=A cos θ, Aᵧ=A sin θ; From vertical: Aₓ=A sin θ, Aᵧ=A cos θ; A=√(Aₓ²+Aᵧ²); θ=tan⁻¹(Aᵧ/Aₓ)
- Example
- 20 N at 30° → Fₓ=17.32 N, Fᵧ=10 N; Aₓ=−6 N, Aᵧ=8 N → A=10 N, θ=126.87°
- Critical warning
- Assuming the angle is always measured from the horizontal.
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Practice
Beginner
Q1. Define vector resolution.
Beginner
Q2. What are rectangular components?
Beginner
Q3. Write the formulas for Aₓ and Aᵧ.
Beginner
Q4. What do î and ĵ represent?
Beginner
Q5. Can a vector component have a negative sign?
Intermediate
Q6. Resolve a force of 40 N acting at 60° above the horizontal.
Intermediate
Q7. Find the components of a velocity of 25 m s⁻¹ acting at 35° from the horizontal.
Intermediate
Q8. Calculate the magnitude of a vector whose components are 5 N and 12 N.
Intermediate
Q9. Find the direction of a vector having components 8 m and 6 m.
Intermediate
Q10. Write A⃗ in unit-vector form if Aₓ = −4 N and Aᵧ = 7 N.
Advanced
Q11. Resolve a 70 N vector making 140° with the positive x-axis.
Advanced
Q12. A vector has components −9 N and −12 N. Find its magnitude and quadrant.
Advanced
Q13. A 10 kg block is placed on a smooth plane inclined at 25°. Find the components of its weight.
Advanced
Q14. A vector has magnitude 100 N and a vertical component of 60 N. Find its positive horizontal component.
Advanced
Q15. Explain how component formulas change when the angle is measured from the vertical.
Exam Practice
Important Short Question
Q1. What is the resolution of a vector?
Important Short Question
Q2. What are rectangular components?
Important Short Question
Q3. Can a vector component be negative?
Important Short Question
Q4. Why are vectors resolved into components?
Long Question Preparation
Q5. Explain the resolution of a vector into rectangular components.