9 min read
Product of Vectors
The multiplication of two vectors according to a defined mathematical rule is called the product of vectors.
Choose Your Level
Product of Vectors
Subtopics
- Scalar Product or Dot Product
- Vector Product or Cross Product
Learning Outcomes
- Explain the meaning of vector multiplication.
- Differentiate between dot product and cross product.
- Calculate the dot product using magnitudes and components.
- Calculate the angle between two vectors using their dot product.
- Calculate the magnitude and direction of a cross product.
- Apply the right-hand rule correctly.
- Calculate a cross product using Cartesian components.
- Apply vector products to work, torque, power, angular momentum, and magnetic force.
Memory Rule Dot → cosine → scalar Cross → sine → vector
Recommended for Class 11
Product of Vectors
Chapter 2 — Vectors · Topic: Product of Vectors Subtopics: 1. Scalar Product or Dot Product 2. Vector Product or Cross Product
Learning Outcomes
- Explain the meaning of vector multiplication.
- Differentiate between dot product and cross product.
- Calculate the dot product using magnitudes and components.
- Calculate the angle between two vectors using their dot product.
- Calculate the magnitude and direction of a cross product.
- Apply the right-hand rule correctly.
- Calculate a cross product using Cartesian components.
- Apply vector products to work, torque, power, angular momentum, and magnetic force.
Definition of Product of Vectors
Dot Product and Cross Product Comparison
Dot product versus cross product
| Feature | Dot product | Cross product |
|---|---|---|
| Symbol | A⃗ · B⃗ | A⃗ × B⃗ |
| Formula | AB cos θ | AB sin θ n̂ |
| Result | Scalar | Vector |
| Direction | No direction | Perpendicular to both vectors |
| Order | Commutative | Anti-commutative |
| Maximum value | Parallel vectors | Perpendicular vectors |
| Zero value | Perpendicular vectors | Parallel vectors |
| Main applications | Work and power | Torque and angular momentum |
Key Logic
Tips and Tricks
Tips and Tricks
- Check whether the required answer is scalar or vector.
- Use cosine for the dot product.
- Use sine for the cross product.
- Use the smaller angle between the vectors.
- Never attach a direction to a dot-product answer.
- Always determine the direction of a cross product.
- Changing the order does not change the dot product.
- Changing the order reverses the cross product.
- Check the signs carefully in component calculations.
- Keep calculator mode in degrees when angles are given in degrees.
Common Mistakes
Common Mistakes
- Using sine in the dot-product formula.
- Using cosine in the cross-product formula.
- Writing a vector as the answer to a dot product.
- Forgetting the direction of a cross product.
- Assuming the cross product is commutative.
- Using the wrong order in unit-vector multiplication.
- Missing the negative sign in the ĵ term of determinant expansion.
- Confusing A⃗ · A⃗ with A⃗ × A⃗.
- Forgetting that perpendicular vectors have zero dot product.
- Forgetting that parallel vectors have zero cross product.
Topic Summary
Vectors can be multiplied using scalar and vector products. The dot product produces a scalar and measures the parallel relationship between vectors. The cross product produces a perpendicular vector and measures the perpendicular interaction between vectors. Correct formulas, component signs, vector order, angles, and direction rules are essential for solving these problems.
Exam Facts
Exam Facts
- The result of a dot product is a scalar.
- The result of a cross product is a vector.
- The dot product is commutative.
- The cross product is anti-commutative.
- Perpendicular vectors have a zero dot product.
- Parallel vectors have a zero cross product.
- Work is an application of the dot product.
- Torque is an application of the cross product.
- The direction of a cross product is found using the right-hand rule.
- The magnitude of a cross product gives the area of a parallelogram.
Exam Focus
Exam Focus
Important points
- The result of a dot product is a scalar.
- The result of a cross product is a vector.
- The dot product is commutative.
- The cross product is anti-commutative.
- Perpendicular vectors have a zero dot product.
- Parallel vectors have a zero cross product.
- Work is an application of the dot product.
- Torque is an application of the cross product.
- The direction of a cross product is found using the right-hand rule.
- The magnitude of a cross product gives the area of a parallelogram.
Important formulas
- A⃗ · B⃗ = AB cos θ
- A⃗ · B⃗ = AₓBₓ + AᵧBᵧ + AzBz
- θ = cos⁻¹((A⃗ · B⃗) / AB)
- W = F⃗ · s⃗ = Fs cos θ
- A⃗ × B⃗ = AB sin θ n̂
- |A⃗ × B⃗| = AB sin θ
- A⃗ × B⃗ = −(B⃗ × A⃗)
- τ⃗ = r⃗ × F⃗
- Area of parallelogram = |A⃗ × B⃗|
Common exam traps
- Using sine in the dot-product formula.
- Using cosine in the cross-product formula.
- Writing a vector as the answer to a dot product.
- Forgetting the direction of a cross product.
- Assuming the cross product is commutative.
- Using the wrong order in unit-vector multiplication.
- Missing the negative sign in the ĵ term of determinant expansion.
- Confusing A⃗ · A⃗ with A⃗ × A⃗.
Short question tips
- Define the vector product.
- When is the dot product zero?
- Why is the cross product anti-commutative?
Long question structure
Two vectors can be multiplied using the dot-product or cross-product rule. The dot product produces a scalar quantity. It is defined as: A⃗ · B⃗ = AB cos θ It measures the parallel interaction of two vectors. The dot product is commutative. Therefore: A⃗ · B⃗ = B⃗ · A⃗ The dot product of perpendicular vectors is zero. Work done is an application of the dot product. The cross product produces a vector quantity. It is defined as: A⃗ × B⃗ = AB sin θ n̂ Its direction is perpendicular to both original vectors. The right-hand rule determines this direction. The cross product is anti-commutative. Therefore: A⃗ × B⃗ = −(B⃗ × A⃗) The cross product of parallel vectors is zero. Its magnitude is maximum for perpendicular vectors. Torque is an application of the cross product. Thus, dot and cross products represent different forms of vector multiplication.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Dot product produces a scalar; uses cosine; commutative; zero at 90°; used in work.
Cross product produces a vector; uses sine; anti-commutative; zero at 0° and 180°; used in torque.
The multiplication of two vectors according to a defined mathematical rule is called the product of vectors.
A⃗·B⃗=AB cos θ = AₓBₓ+AᵧBᵧ+AzBz; A⃗×B⃗=AB sin θ n̂ = −(B⃗×A⃗)
- Definition
- The multiplication of two vectors according to a defined mathematical rule is called the product of vectors.
- Formula
- A⃗·B⃗=AB cos θ = AₓBₓ+AᵧBᵧ+AzBz; A⃗×B⃗=AB sin θ n̂ = −(B⃗×A⃗)
- Example
- 6 and 8 at 60° → A⃗·B⃗=24; |A⃗×B⃗| for 5 and 7 at 30° = 17.5; W=100 J; τ=20 N m
- Critical warning
- Using sine in the dot-product formula.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Practice
Beginner
Q1. State two differences between dot and cross products.
Exam Practice
Long Question Preparation
Q1. Explain the dot product and cross product of two vectors.