Cross Product Linearity

Author:
Vincent Tam
Topic:
Vectors 3D (Three-Dimensional), Algebra, Coordinates, Geometry

Goal

Recall

  1. componentwise addition: , and
  2. linearity of dot products: + +
  3. dot product and parallel component:
  4. cross product and perpendicular component:

Link

Key: take away the parallel components and parallel to , since they don't "rotate ".
  • LHS:
  • RHS:

Observation

With the vector pointing upwards, the vector is
  1. rotated 90° anticlockwise, then
  2. scaled by a factor of
to form the vector . Both actions don't change the shape of the parallelogram (with as the diagonal), so .

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