In 3D, a cross product takes two vectors and returns a vector. Geometrically, the magnitude of the result equals to parallelogram area and the direction is orthogonal to both vectors. cross_product.png

Properties

  1. is orthogonal to both and
  2. The vectors , , are right handed
  3. (anti-commutativity)
  4. and (distributivity)
  5. (associativity)

Cross Product, Determinant, and Angle

We can have a precise definition of the cross product without the right hand rule by using determinant:

Formula: cross_product_as_determinant.png

Cross Product as Quarter Rotation

cross_product_as_quarter_rotation.png

An observation for manipulating vectors in 3D: cross product with a unit vector is equivalent to quarter-rotation in the plane with normal : This insight is important for axis-angle rotation, as we can using and as basis vectors and use trig to find arbitrary rotation of around .

See Also

Reference