Study Notes on Cross Product and Right Hand Rule

Cross Product and Right Hand Rule

Overview

• Discussion on the cross product of vectors and the directions associated with it.

Directionality of Cross Product

• Two perpendicular directions to the plane of vectors.
  • One direction is upward.
  • The other direction is downward.
• Property 1: There are only two possible directions of the cross product.
  • Determination of the correct direction will be discussed later.

Vector Representation

• Definition of vectors using colors:
  • Red marker represents vector u.
  • Green marker represents vector v.

The Right Hand Rule

• Definition of the Right Hand Rule: A method to determine the direction of the cross product using the right hand.
• The sequence of application: U cross V, which specifies the order of vectors (first u, then v).

Application of the Right Hand Rule

1. Align the right hand to vector u (red marker).
   • Two possible alignments for fingers relative to vector u.
   • Alternative hand orientations considering the direction of v.
2. Choose the alignment that allows fingers to comfortably point toward vector v.
   • Thumb will indicate the direction of the cross product.
3. If the wrong alignment is chosen:
   • Inability to move fingers toward v indicates incorrect hand position.

Mathematical Aspects of Cross Product

• The magnitude of the cross product is given by: $|u imes v| = |u| |v| imes ext{sine}( heta)$
  • The importance of θ (angle between the two vectors) in defining the cross product's magnitude.
  • Future discussions will cover the geometric interpretation of this formula.

Examples of Vectors on the XY Plane

• Consider vectors u and v in the XY plane.
• Observation when changing the order of vectors: u and v to v and u.

Finding V cross U using the Right Hand Rule

1. Align fingers with vector v (green marker).
2. Move fingers to point toward vector u (red marker).
3. Resulting direction of V cross U:
   • This will be downward.
   • Confirmed that vector v points inward and hand alignment indicates down is the correct direction.

Conclusion on Order of Cross Products

• Important property derived:
  • Changing the order of cross product yields opposite directions.
• Final question addressed: What is the cross product of a vector with itself?
  • The angle θ between a vector and itself is 0 degrees.
  • Therefore, the cross product's magnitude is 0:
    $|u imes u| = 0$

Summary of Key Points

• The right hand rule is a fundamental technique for determining the direction of the cross product.
• Vectors u and v yield different cross products based on their order:
  $u imes v$ points upwards, while $v imes u$ points downwards.
• The cross product of a vector with itself is always zero due to the angle being zero between the same vectors.