Calculating a Vector Product

Calculating a Vector Product

Basics of Vector Products

  • A vector is an object with both magnitude (size) and direction.
  • The vector product (or cross product) is a binary operation on two vectors in three-dimensional space.
  • The cross product of vectors A and B is written as A × B.

Properties of the Vector Product

  • The vector product A × B gives a vector which is perpendicular (at a right angle) to both A and B.
  • The magnitude (length) of A × B is equal to the area of the parallelogram with A and B as sides.
  • The direction of the vector product is given by the right-hand rule. If the fingers of the right hand are curled from the first vector A to the second vector B, the thumb gives the direction of A × B.

Calculation of the Vector Product

  • In Cartesian coordinates, if A = (A1, A2, A3) and B = (B1, B2, B3), then the cross product A × B can be found by calculating the determinant of following matrix:

    | i  j  k |
| A1 A2 A3|
| B1 B2 B3|

  • Expand the determinant to find the i, j, and k (or x, y, z) components of the cross product.

Misc Properties of Vector Products

  • The cross product is not commutative - A × B ≠ B × A.
  • The cross product is distributive over addition - A × (B + C) = A × B + A × C.
  • There is not a general associative law for the cross product - A × (B × C) ≠ (A × B) × C.

Application of the Vector Product

  • Vector products are useful in physics and engineering to describe rotational motion and to compute torque.
  • In computer graphics, vector products are used to compute lighting conditions and render 3D models.

Note

Remember to always use the right-hand rule to find the direction of the cross product and the rules for this operation with vectors.