Cross Product of Vectors

Introduction

• Topic: Cross Product of Vectors (Vector Product)
• Previous topic studied: Dot Product

Greeting

• "Very good evening, good morning, good afternoon, or good night" depending on the time of day.

Concept of Cross Product

• When multiplying two vectors A and B with a cross sign (A × B), the result is also a vector.
• This is known as the cross product, distinct from the dot product where the result is a scalar.

Coordinate System & Torque

• Cross products are essential in understanding phenomena such as torque and moment of force.
• Torque formula: Torque = Force × Displacement

Formula for Cross Product

• The magnitude of the cross product can be calculated as:

  |A × B| = |A| |B| sin(θ)

  where θ is the angle between vectors A and B.

• Direction of cross product (C = A × B) is given by a unit vector (n̂):

  n̂ = perpendicular to both A and B

• The unit vector can point either upward or downward relative to the plane formed by A and B.

Example Calculation

• Given:

  • Magnitude of A = 5
  • Magnitude of B = 2
  • Angle θ = 30°

• Calculate A × B:

  |A × B| = |A| |B| sin(30°) = 5 * 2 * 1/2 = 5

• Calculating B × A gives the same magnitude but different direction due to properties of cross product.

Properties of Cross Product

• Non-commutativity: A × B ≠ B × A
  • A × B may produce a vector pointing upwards, whereas B × A points downwards.
• Right Hand Rule:
  • For direction, use the right hand. Curl fingers from A to B, the thumb points in the direction of A × B.

Orthogonal Unit Vectors

• î, ĵ, k̂ are the standard unit vectors.
• Examples of cross products:
  • î × ĵ = k̂
  • ĵ × k̂ = î
  • k̂ × î = ĵ
• Self cross products:
  • î × î = 0, ĵ × ĵ = 0, k̂ × k̂ = 0

Cross Product Calculation Using Determinants

• Use the determinant method to calculate cross products:
  • Matrix setup:
    | î ĵ k̂ |
    | a₁ a₂ a₃ |
    | b₁ b₂ b₃ |
• Calculate as shown previously:
  • A × B will yield a new vector based on established unit vectors.

Key Point Summary

• A × B = 0 if:
  • A and B are parallel (0° or 180°)
• A ⋅ B = 0 means A and B are perpendicular (90°)

Sample Problems

• Determining angles based on conditions provided (dot and cross products)
• Magnitude checks and problem-solving with given vectors.

Advanced Concepts

• Understanding torque in vector forms: R × F where R is the position vector.
• R is determined by the coordinates of the points involved.

Conclusion

• Practice determining angles, magnitudes, and direction with additional examples in upcoming sessions.
• The next lessons will focus on further advanced vector topics and kinematics.